This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cbvaldw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvaldw.1	$⊢ Ⅎ y φ$
	cbvaldw.2	$⊢ φ \to Ⅎ y ψ$
	cbvaldw.3	$⊢ φ \to (x = y \to (ψ \leftrightarrow χ))$
Assertion	cbvaldw	$⊢ φ \to (\forall x ψ \leftrightarrow \forall y χ)$

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	$⊢ Ⅎ y φ$
2		cbvaldw.2	$⊢ φ \to Ⅎ y ψ$
3		cbvaldw.3	$⊢ φ \to (x = y \to (ψ \leftrightarrow χ))$
4		nfv	$⊢ Ⅎ x φ$
5		nfvd	$⊢ φ \to Ⅎ x χ$
6	4 1 2 5 3	cbv2w	$⊢ φ \to (\forall x ψ \leftrightarrow \forall y χ)$