Theorem cbval2 1914

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)

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 1569	. 2 ⊢ Ⅎ𝑧∀𝑦𝜑
3		cbval2.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfal 1569	. 2 ⊢ Ⅎ𝑥∀𝑤𝜓
5		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧
6		cbval2.2	. . . . . 6 ⊢ Ⅎ𝑤𝜑
7	5, 6	nfim 1565	. . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 → 𝜑)
8		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧
9		cbval2.4	. . . . . 6 ⊢ Ⅎ𝑦𝜓
10	8, 9	nfim 1565	. . . . 5 ⊢ Ⅎ𝑦(𝑥 = 𝑧 → 𝜓)
11		cbval2.5	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
12	11	expcom 115	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)))
13	12	pm5.74d 181	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜓)))
14	7, 10, 13	cbval 1747	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑤(𝑥 = 𝑧 → 𝜓))
15		19.21v 1866	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑))
16		19.21v 1866	. . . 4 ⊢ (∀𝑤(𝑥 = 𝑧 → 𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
17	14, 15, 16	3bitr3i 209	. . 3 ⊢ ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
18	17	pm5.74ri 180	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
19	2, 4, 18	cbval 1747	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  cbval2v  1916