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 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvsbcvw	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑦 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvsbcw	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑦 ] 𝜓 )