Help in proving a tautology

Question

I am having real trouble deriving this tautology:

$\forall(x) ((x=a) \lor (x\neq a))$

It is easy to solve this by assuming the negation, unpack the negation with DeMorgan's Law, and derive from there.

I am not allowed to use DeMorgan's however. Because of this I am at loss.

Any help will be greatly appreciated. Thank you!

Answer 1

It's not clear in your question what are we allowed to do, anyway here is a hint:

First note that $x \neq a$ is nothing but $\neg x = a$, hence we should prove that:

$$\vdash \forall x (x=a \vee \neg x=a)$$

Now recall that

$\alpha \rightarrow \beta \equiv \alpha \vee \neg \beta$

Now isn't there an implicational tautology you could easily prove instead of $x=a \vee \neg x=a$?

Answer 2

Let's call the first statement "$x=a$" : $A$

and the second statement "$x \neq a$ : $B$

So clearly, $ B = A'$

Hence, $A \vee B = A \vee A' = true$