Theorem cbvex2 1910

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

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
cbvex2	⊢ (∃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 cbvex2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfex 1625	. 2 ⊢ Ⅎ𝑧∃𝑦𝜑
3		cbval2.3	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfex 1625	. 2 ⊢ Ⅎ𝑥∃𝑤𝜓
5		nfv 1516	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 = 𝑧
6		cbval2.2	. . . . . 6 ⊢ Ⅎ𝑤𝜑
7	5, 6	nfan 1553	. . . . 5 ⊢ Ⅎ𝑤(𝑥 = 𝑧 ∧ 𝜑)
8		nfv 1516	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝑧
9		cbval2.4	. . . . . 6 ⊢ Ⅎ𝑦𝜓
10	8, 9	nfan 1553	. . . . 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	pm5.32d 446	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑥 = 𝑧 ∧ 𝜓)))
14	7, 10, 13	cbvex 1744	. . . 4 ⊢ (∃𝑦(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑤(𝑥 = 𝑧 ∧ 𝜓))
15		19.42v 1894	. . . 4 ⊢ (∃𝑦(𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦𝜑))
16		19.42v 1894	. . . 4 ⊢ (∃𝑤(𝑥 = 𝑧 ∧ 𝜓) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
17	14, 15, 16	3bitr3i 209	. . 3 ⊢ ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓))
18		pm5.32 449	. . 3 ⊢ ((𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) ↔ ((𝑥 = 𝑧 ∧ ∃𝑦𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑤𝜓)))
19	17, 18	mpbir 145	. 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
20	2, 4, 19	cbvex 1744	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1448  ∃wex 1480

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  cbvex2v  1912  cbvopab  4053  cbvoprab12  5916