First order logic deduction

Question

I have the following formula.

$\forall X (X=A \Rightarrow x \in X)$

I think this just means that $x \in A$ (right?), but I can't provide a formal proof. Can anyone help me understand which tools I should be using?

Answer 1

Here is a formal proof along the lines as suggested by @TannerSwett:

Answer 2

I think this just means that $x \in A$ (right?), but I can't provide a formal proof.

That's correct. So the statement that you need to prove is:

$$\forall X (X = A \implies x \in X) \iff x \in A$$

The way to prove this is by proving that $\forall X (X = A \implies x \in X) \implies x \in A$, and also proving that $x \in A \implies \forall X (X = A \implies x \in X)$.

Forward direction

You may assume that $\forall X (X = A \implies x \in X)$, and you must prove that $x \in A$. Use universal instantiation here.

Backward direction

You may assume that $x \in A$, and you must prove that $\forall X (X = A \implies x \in X)$. Start by proving that $X = A \implies x \in X$, and then use universal generalization.

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I hope this provides a step in the right direction. How far can you go from here? Do you have any questions about this?