Theorem cbvrmov

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Alexander van der Vekens, 17-Jun-2017) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvrmov.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrmov	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvrmov.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbvrmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$