Theorem cbvsbcv

Description: Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvsbcvw when possible. (Contributed by NM, 30-Sep-2008) (Revised by Mario Carneiro, 13-Oct-2016) (New usage is discouraged.)

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

Proof

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