Theorem cbval2 2411

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbval2v 2342 if possible. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbval2.1	⊢ Ⅎ𝑧𝜑
cbval2.2	⊢ Ⅎ𝑤𝜑
cbval2.3	⊢ Ⅎ𝑥𝜓
cbval2.4	⊢ Ⅎ𝑦𝜓
cbval2.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbval2	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧   𝑥,𝑤   𝑧,𝑤

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

Proof of Theorem cbval2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfal 2321	. 2 ⊢ Ⅎ𝑧∀𝑦𝜑
3		cbval2.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2321	. 2 ⊢ Ⅎ𝑥∀𝑤𝜓
5		nfv 1918	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝑧
6		nfv 1918	. . 3 ⊢ Ⅎ𝑤 𝑥 = 𝑧
7		cbval2.2	. . . 4 ⊢ Ⅎ𝑤𝜑
8	7	a1i 11	. . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑤𝜑)
9		cbval2.4	. . . 4 ⊢ Ⅎ𝑦𝜓
10	9	a1i 11	. . 3 ⊢ (𝑥 = 𝑧 → Ⅎ𝑦𝜓)
11		cbval2.5	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
12	11	ex 412	. . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑤 → (𝜑 ↔ 𝜓)))
13	5, 6, 8, 10, 12	cbv2 2403	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
14	2, 4, 13	cbval 2398	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  cbvex2  2412