Theorem cbvex2v 2340

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2410 with a disjoint variable condition, which does not require ax-13 2370. (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 1858	. . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑
3		cbval2v.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	nfn 1858	. . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑
5		cbval2v.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1858	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbval2v.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	7	nfn 1858	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓
9		cbval2v.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
10	9	notbid 318	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
11	2, 4, 6, 8, 10	cbval2v 2338	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
12		2nexaln 1830	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
13		2nexaln 1830	. . 3 ⊢ (¬ ∃𝑧∃𝑤𝜓 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
14	11, 12, 13	3bitr4i 303	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑧∃𝑤𝜓)
15	14	con4bii 321	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1537  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-nf 1784

This theorem is referenced by:  cbvopab  5153  cbvoprab12  7396  bj-cbvex2vv  35033  or2expropbilem2  44771  ichnreuop  45168