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Metamath Proof Explorer

Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by GG, 23-May-2024)

	Ref	Expression
Hypotheses	cbvabw.1	$⊢ Ⅎ y φ$
	cbvabw.2	$⊢ Ⅎ x ψ$
	cbvabw.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	cbvabw	$⊢ \{x \| φ\} = \{y \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		cbvabw.1	$⊢ Ⅎ y φ$
2		cbvabw.2	$⊢ Ⅎ x ψ$
3		cbvabw.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
4	1 2 3	cbvsbvf	$⊢ [z/ x] φ \leftrightarrow [z/ y] ψ$
5		df-clab	$⊢ z \in \{x \| φ\} \leftrightarrow [z/ x] φ$
6		df-clab	$⊢ z \in \{y \| ψ\} \leftrightarrow [z/ y] ψ$
7	4 5 6	3bitr4i	$⊢ z \in \{x \| φ\} \leftrightarrow z \in \{y \| ψ\}$
8	7	eqriv	$⊢ \{x \| φ\} = \{y \| ψ\}$