Given:  nothing
Derive:
((P==>R) & (Q==>R)) <==> ((P v Q) ==> R)

7.  If I know P==>Q and I know P, I can deduce Q.
8.  If, under the assumption of P, I can derive Q, then I can deduce P==>Q

9.  If I know P<==>Q and I know P, I can deduce Q.  If I know P <==> Q and I know Q, I can deduce P.

10.  If, under the assumption of P, I can derive Q, AND under the assumption of Q, I can derive P, then I can deduce P <==> Q.

1.  ((P==>R) & (Q==>R)) <==> ((P v Q) ==> R)
    ________________________________________
2.	Assume (P==>R) & (Q==>R)
	_________________________
3.		Assume P v Q
		____________
4.		(P==>R) & (Q==>R) (Assumption 2)
5.              P==>R (4)
6.              Q==>R (5)
7.			Assume P
			________
8.			P (Assumption 7)
9.			R (4,8)

10.			Assume Q
			________
11.			Q (10)
			R (6,11)
12.		P v Q (Assumption 3)
13.		R (12,7-9,10-12)
14.	(P v Q) ==> R (3-13)

15.	Assume ((P v Q) ==> R)
	______________________
16.	((P v Q) ==> R) (Assumption 15)
17.		Assume P
		________
18.		P (Assumption 17)
19.		P v Q (18)
20.		R (16,19)

21.	P ==> R (17-20)

22.		Assume Q
		________
23.		Q (Assumption 22)
24.		P v Q (23)
25.		R (16,24)
26.	Q ==> R (22-25)
27.	(P ==> R) & (Q ==> R) (21,26)

28.  ((P==>R) & (Q==>R)) <==> ((P v Q) ==> R) (2-14, 15-27)

Cube Root of 1:

1

-1/2 + sqrt(3)/2 i

-1/2 - sqrt(3)/2 i

Quantificational Logic (or first order logic).

You imagine that you have a set of "stuff" and this set may well be infinite.  This is your "universe."

In addition to "predicates" like P or Q, you have propositions that apply to elements in your universe.  Px or Qxy.  Px means this property holds specifically for element x.  Qxy means that Q is true as applied to x and y in that order.  Qxy and Qyx may mean entirely different things.  Qxx is also a valid statement.

We still have a binary going on.  Statements are true or false, but it's like we can pass "parameters" to them now.

Let's say my universe is past and present U.S. Presidents.

Px means "x is female."

There is no value that makes this true.

Qx means "x is proudly bald."

What x makes this true?  x might be "Van Buren".  x might be "Eisenhower."

Rxy means "x is the father of y."

x = "John Adams" y = "John Quincy Adams"

x = "George H.W. Bush" y = "George W. Bush"

Sxy = means "x is the ancestor of y."

x = "William Henry Harrison" y = "Benjamin Harrison"

So, there are major fundamental logical operations (aka quantifiers) that apply to these sorts of propositions.

The existential quantifier:  There exists

(Ex):  There exists an x.  The real E is upside down (rotated on the y or z-axes).  Not in my text file it's not, but in actual books it is.

The universal quantifier:  For all.

(Ax):  For all x.  The real A is upside down (rotated on either the x or z-axes.

(Ax)Px:  this means that every single x in the universe makes Px true.

(Ex)Px:  this means that there is something (possibly more than one something) but at least one something that makes Px true.

Free vs. bound variables.

A bound variable is a variable that is tied to a quantifier.  So, if I say (Ax)Px, then the x is bound.  This means that x doesn't refer to any particular or specific thing in the universe, it's part of a broader statement about the entire universe.  In other words, (Ax)Px doesn't have a value of true or false depending on any specific value of x.  The statement is either completely or completely false.

Px is "x is even."
P(12) is true.
P(11) is false.
(Ax)Px is false without specifying any specific x.

Px:  The x in this statement is free.  Px may be true or false depending on what that x actually is.

If S is a statement, S (x/y) means that FREE appearances of y in S are replaced with x.

S = ((Ay)Py) & (Ex)Qxy

S(x/y) = ((Ay)Py) & (Ex)Qxx

13.  If I know (Ax)Px then I can derive P(y/x).  y can be a specific thing in the universe or a variable, provided that y is free in P.

"If I know that something is ALWAYS true, then it must be true for any specific example I need."

14.  If I know Pu and u does not appear free in P or in any given statements or assumptions, then I can deduce (Ax)Px.

"If I can prove P to be true for one general example, not using any specific properties of that example, then P must always be true."