Theorem cbvsbcvw

Description: Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 30-Sep-2008) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbvsbcvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvsbcvw	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvsbcvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbvsbcw	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\psi$$