Theorem bj-cbv3hv2

Description: Version of cbv3h with two disjoint variable conditions, which does not require ax-11 nor ax-13 . (Contributed by BJ, 24-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbv3hv2.nf	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	bj-cbv3hv2.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	bj-cbv3hv2	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbv3hv2.nf	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		bj-cbv3hv2.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1	nf5i	⊢ Ⅎ 𝑥 𝜓
4	3 2	cbv3v2	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )