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	1	cbvabv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$
3	2	eleq2i	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow A\in \left\{y|\psi \right\}$$
4		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{y|\psi \right\}$$
6	3 4 5	3bitr4i	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\psi$$