Theorem cbvex2v 2379

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2447 with a disjoint variable condition, which does not require ax-13 2407. (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 1890	. . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑
3		cbval2v.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	nfn 1890	. . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑
5		cbval2v.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1890	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbval2v.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	7	nfn 1890	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓
9		cbval2v.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
10	9	notbid 321	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
11	2, 4, 6, 8, 10	cbval2v 2378	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
12		2nexaln 1863	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
13		2nexaln 1863	. . 3 ⊢ (¬ ∃𝑧∃𝑤𝜓 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
14	11, 12, 13	3bitr4i 306	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑧∃𝑤𝜓)
15	14	con4bii 324	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  cbvopab  5188  cbvoprab12  7512  bj-cbvex2vv  37478  or2expropbilem2  47811  ichnreuop  48262