Theorem cbvrexv

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexvw when possible. (Contributed by NM, 2-Jun-1998) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvralv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrexv	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbvrex	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$