Theorem bj-cbvaldv

Description: Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvaldv.1	⊢ Ⅎ 𝑦 𝜑
	bj-cbvaldv.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	bj-cbvaldv.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	bj-cbvaldv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvaldv.1	⊢ Ⅎ 𝑦 𝜑
2		bj-cbvaldv.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		bj-cbvaldv.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4		nfv	⊢ Ⅎ 𝑥 𝜑
5		nfv	⊢ Ⅎ 𝑥 𝜒
6	5	a1i	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
7	4 1 2 6 3	bj-cbv2v	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )