Theorem cbval2v 2377

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2 2445 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbval2v.1	⊢ Ⅎ𝑧𝜑
cbval2v.2	⊢ Ⅎ𝑤𝜑
cbval2v.3	⊢ Ⅎ𝑥𝜓
cbval2v.4	⊢ Ⅎ𝑦𝜓
cbval2v.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbval2v	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbval2v

Step	Hyp	Ref	Expression
1		cbval2v.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfal 2358	. 2 ⊢ Ⅎ𝑧∀𝑦𝜑
3		cbval2v.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2358	. 2 ⊢ Ⅎ𝑥∀𝑤𝜓
5		nfv 1937	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝑧
6		nfv 1937	. . 3 ⊢ Ⅎ𝑤 𝑥 = 𝑧
7		cbval2v.2	. . . 4 ⊢ Ⅎ𝑤𝜑
8	7	a1i 11	. . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑤𝜑)
9		cbval2v.4	. . . 4 ⊢ Ⅎ𝑦𝜓
10	9	a1i 11	. . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑦𝜓)
11		cbval2v.5	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
12	11	ex 417	. . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑤 → (𝜑 ↔ 𝜓)))
13	5, 6, 8, 10, 12	cbv2w 2371	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
14	2, 4, 13	cbvalv1 2375	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by:  cbvex2v  2378  bj-cbval2vv  37293  eqrelf  38764