Theorem cbv2OLD 2426

Description: Obsolete version of cbv2 2422 as of 10-Sep-2023. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
cbv2OLD.1	⊢ Ⅎ𝑥𝜑
cbv2OLD.2	⊢ Ⅎ𝑦𝜑
cbv2OLD.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv2OLD.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv2OLD.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2OLD	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2OLD

Step	Hyp	Ref	Expression
1		cbv2OLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbv2OLD.2	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	nf5ri 2194	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4	1, 3	alrimi 2212	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
5		cbv2OLD.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
6	5	nf5rd 2195	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
7		cbv2OLD.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
8	7	nf5rd 2195	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
9		cbv2OLD.5	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
10	6, 8, 9	cbv2h 2425	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
11	4, 10	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)