Theorem expandexn

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

	Ref	Expression
Hypothesis	expandexn.1	$$⊢\phi \leftrightarrow \neg \psi$$
Assertion	expandexn	$$⊢\exists x\phi \leftrightarrow \neg \forall x\psi$$

Proof

Step	Hyp	Ref	Expression
1		expandexn.1	$$⊢\phi \leftrightarrow \neg \psi$$
2	1	exbii	$$⊢\exists x\phi \leftrightarrow \exists x\neg \psi$$
3		exnal	$$⊢\exists x\neg \psi \leftrightarrow \neg \forall x\psi$$
4	2 3	bitri	$$⊢\exists x\phi \leftrightarrow \neg \forall x\psi$$