Theorem cbv2 2424

Description: Rule used to change bound variables, using implicit substitution. (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.)

Hypotheses

Ref	Expression
cbv2.1	⊢ Ⅎ𝑥𝜑
cbv2.2	⊢ Ⅎ𝑦𝜑
cbv2.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv2.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv2.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbv2.2	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	nf5ri 2238	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4	1, 3	alrimi 2258	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑)
5		cbv2.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
6	5	nf5rd 2240	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
7		cbv2.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
8	7	nf5rd 2240	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
9		cbv2.5	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
10	6, 8, 9	cbv2h 2423	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
11	4, 10	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1656  Ⅎwnf 1884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885

This theorem is referenced by:  cbvald  2429  sb9  2559  wl-cbvalnaed  33864  wl-sb8t  33879