Theorem bj-cbv2hv 37542

Description: Version of cbv2h 2437 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv2hv.1	⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
bj-cbv2hv.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
bj-cbv2hv.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
bj-cbv2hv	⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)

Proof of Theorem bj-cbv2hv

Step	Hyp	Ref	Expression
1		bj-cbv2hv.1	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
2		bj-cbv2hv.2	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
3		bj-cbv2hv.3	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
4		biimp 218	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
5	3, 4	syl6 36	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	1, 2, 5	bj-cbv1hv 37541	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7		equcomi 2050	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		biimpr 223	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
9	7, 3, 8	syl56 37	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓)))
10	2, 1, 9	bj-cbv1hv 37541	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
11	10	alcoms 2195	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
12	6, 11	impbid 215	1 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  bj-cbv2v  37543