Theorem expandrexn

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

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

Proof

Step	Hyp	Ref	Expression
1		expandrexn.1	$$⊢\phi \leftrightarrow \neg \psi$$
2	1	rexbii	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in A\neg \psi$$
3		df-rex	$$⊢\exists x\in A\neg \psi \leftrightarrow \exists x\left(x\in A\wedge \neg \psi \right)$$
4		exanali	$$⊢\exists x\left(x\in A\wedge \neg \psi \right)\leftrightarrow \neg \forall x\left(x\in A\to \psi \right)$$
5	2 3 4	3bitri	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\left(x\in A\to \psi \right)$$