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Metamath Proof Explorer

Theorem cbv3v2

Description: Version of cbv3 with two disjoint variable conditions, which does not require ax-11 nor ax-13 . (Contributed by BJ, 24-Jun-2019) (Proof shortened by Wolf Lammen, 30-Aug-2021)

	Ref	Expression
Hypotheses	cbv3v2.nf	$⊢ Ⅎ x ψ$
	cbv3v2.1	$⊢ x = y \to (φ \to ψ)$
Assertion	cbv3v2	$⊢ \forall x φ \to \forall y ψ$

Proof

Step	Hyp	Ref	Expression
1		cbv3v2.nf	$⊢ Ⅎ x ψ$
2		cbv3v2.1	$⊢ x = y \to (φ \to ψ)$
3	1 2	spimfv	$⊢ \forall x φ \to ψ$
4	3	alrimiv	$⊢ \forall x φ \to \forall y ψ$