Theorem expandrex

Description: Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypothesis	expandrex.1	$$⊢\phi \leftrightarrow \psi$$
Assertion	expandrex	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\left(x\in A\to \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		expandrex.1	$$⊢\phi \leftrightarrow \psi$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1 2	bitri	$$⊢\phi \leftrightarrow \neg \neg \psi$$
4	3	expandrexn	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\left(x\in A\to \neg \psi \right)$$