Theorem cbv2OLD 2416

Description: Obsolete version of cbv2 2412 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 2193	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4	1, 3	alrimi 2211	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
5		cbv2OLD.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
6	5	nf5rd 2194	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
7		cbv2OLD.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
8	7	nf5rd 2194	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
9		cbv2OLD.5	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
10	6, 8, 9	cbv2h 2415	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
11	4, 10	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)