Theorem cbvrexv2

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by David Moews, 1-May-2017) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvralv2.1	$$⊢x=y\to \left(\psi \leftrightarrow \chi \right)$$
	cbvralv2.2	$$⊢x=y\to A=B$$
Assertion	cbvrexv2	$$⊢\exists x\in A\psi \leftrightarrow \exists y\in B\chi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralv2.1	$$⊢x=y\to \left(\psi \leftrightarrow \chi \right)$$
2		cbvralv2.2	$$⊢x=y\to A=B$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xB$$
5		nfv	$$⊢Ⅎy\psi$$
6		nfv	$$⊢Ⅎx\chi$$
7	3 4 5 6 2 1	cbvrexcsf	$$⊢\exists x\in A\psi \leftrightarrow \exists y\in B\chi$$