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

Metamath Proof Explorer

Theorem cbv3v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbv3v.nf1	$⊢ Ⅎ y φ$
	cbv3v.nf2	$⊢ Ⅎ x ψ$
	cbv3v.1	$⊢ x = y \to (φ \to ψ)$
Assertion	cbv3v	$⊢ \forall x φ \to \forall y ψ$

Proof

Step	Hyp	Ref	Expression
1		cbv3v.nf1	$⊢ Ⅎ y φ$
2		cbv3v.nf2	$⊢ Ⅎ x ψ$
3		cbv3v.1	$⊢ x = y \to (φ \to ψ)$
4	1	nf5ri	$⊢ φ \to \forall y φ$
5	4	hbal	$⊢ \forall x φ \to \forall y \forall x φ$
6	2 3	spimfv	$⊢ \forall x φ \to ψ$
7	5 6	alrimih	$⊢ \forall x φ \to \forall y ψ$