Theorem bj-cbv2v 34115

Description: Version of cbv2 2419 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv2v.1	⊢ Ⅎ𝑥𝜑
bj-cbv2v.2	⊢ Ⅎ𝑦𝜑
bj-cbv2v.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
bj-cbv2v.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
bj-cbv2v.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
bj-cbv2v	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-cbv2v

Step	Hyp	Ref	Expression
1		bj-cbv2v.2	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nf5ri 2190	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbv2v.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	nfal 2338	. . . 4 ⊢ Ⅎ𝑥∀𝑦𝜑
5	4	nf5ri 2190	. . 3 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
6	2, 5	syl 17	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
7		bj-cbv2v.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
8	7	nf5rd 2191	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
9		bj-cbv2v.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
10	9	nf5rd 2191	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
11		bj-cbv2v.5	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
12	8, 10, 11	bj-cbv2hv 34114	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
13	6, 12	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2156  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781

This theorem is referenced by:  bj-cbvaldv  34116