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) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

		Ref	Expression
	Hypothesis	cbvsbcvw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	cbvsbcvw	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} A / y \underset{˙}{]} ψ$

Proof

Step	Hyp	Ref	Expression
1		cbvsbcvw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2	1	cbvabv	$⊢ \{x \| φ\} = \{y \| ψ\}$
3	2	eleq2i	$⊢ A \in \{x \| φ\} \leftrightarrow A \in \{y \| ψ\}$
4		df-sbc	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow A \in \{x \| φ\}$
5		df-sbc	$⊢ \underset{˙}{[} A / y \underset{˙}{]} ψ \leftrightarrow A \in \{y \| ψ\}$
6	3 4 5	3bitr4i	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} A / y \underset{˙}{]} ψ$

Metamath Proof Explorer

Theorem cbvsbcvw

Proof