Theorem cbvexvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypothesis	cbvalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvexvw	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	cbvalvw	$$⊢\forall x\neg \phi \leftrightarrow \forall y\neg \psi$$
4	3	notbii	$$⊢\neg \forall x\neg \phi \leftrightarrow \neg \forall y\neg \psi$$
5		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
6		df-ex	$$⊢\exists y\psi \leftrightarrow \neg \forall y\neg \psi$$
7	4 5 6	3bitr4i	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$