Theorem bj-cbv2hv

Description: Version of cbv2h 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-cbv2hv.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
	bj-cbv2hv.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
	bj-cbv2hv.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	bj-cbv2hv	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbv2hv.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
2		bj-cbv2hv.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3		bj-cbv2hv.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4		biimp	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜓 → 𝜒 ) )
5	3 4	syl6	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
6	1 2 5	bj-cbv1hv	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
7		equcomi	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
8		biimpr	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜒 → 𝜓 ) )
9	7 3 8	syl56	⊢ ( 𝜑 → ( 𝑦 = 𝑥 → ( 𝜒 → 𝜓 ) ) )
10	2 1 9	bj-cbv1hv	⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
11	10	alcoms	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
12	6 11	impbid	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )