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 → ( 𝜑 → 𝜓 ) )
Assertion	cbv3v2	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbv3v2.nf	⊢ Ⅎ 𝑥 𝜓
2		cbv3v2.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1 2	spimfv	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
4	3	alrimiv	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )