Theorem cbvrex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexw when possible. (Contributed by NM, 31-Jul-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvral.1	$$⊢Ⅎy\phi$$
	cbvral.2	$$⊢Ⅎx\psi$$
	cbvral.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrex	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvral.1	$$⊢Ⅎy\phi$$
2		cbvral.2	$$⊢Ⅎx\psi$$
3		cbvral.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
6	4 5 1 2 3	cbvrexf	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$