Can someone check if my working is correct for the following proof?

Prove for all sets and .

Let be the proposition function '' and be the proposition function ''

Let the domain of discourse of both these proposition functions be , the universal set.

Now to prove the statement to be true we must show the following 2 universally quantified statements to be true:

1.

2.

To show case 1 to be true, first we know that if is false then the case is vacuously true, so we will ignore this trivial case.

We will assume to be true and if we can show that is also true then case 1 is true.

Now and

Since is true then either

a. is true and is false

b. is false and is true

[Note: these 2 propositions can not be both true since they are mutually exclusive]

If we use case a. Since is true is true is true.

Now using case b. Since is true is true is true is true.

Thus we have shown case 1. to be true.

To show case 2 to be true, first we know that if is false then the case is vacuously true, so again we will ignore this trivial case.

We will assume Q(x) to be true and Q(x) is true if either of the following 3 conditions are satisfied.

i. is true and is false

ii. is false and is true

iii. is true and is true

Now using case i. Since is true is true is true.

Using case ii. Since is true is true is true is true.

Now using case iii. Since is true and is true is true is true.

Case 2 is now proven to be true.

We have completed our proof to show that the original statement is true.

Thanks!