Theorem cbvex2v 2345

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2413 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.)

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
cbvex2v	⊢ (∃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 cbvex2v

Step	Hyp	Ref	Expression
1		cbval2v.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
2	1	nfn 1863	. . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑
3		cbval2v.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	nfn 1863	. . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑
5		cbval2v.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1863	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbval2v.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	7	nfn 1863	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓
9		cbval2v.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
10	9	notbid 317	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
11	2, 4, 6, 8, 10	cbval2v 2343	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
12		2nexaln 1835	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
13		2nexaln 1835	. . 3 ⊢ (¬ ∃𝑧∃𝑤𝜓 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
14	11, 12, 13	3bitr4i 302	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑧∃𝑤𝜓)
15	14	con4bii 320	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1539  ∃wex 1785  Ⅎwnf 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-11 2157  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790

This theorem is referenced by:  cbvopab  5150  cbvoprab12  7355  bj-cbvex2vv  34963  or2expropbilem2  44478  ichnreuop  44876