Theorem cbvex2 2428

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2384. Use the weaker cbvex2v 2359 if possible. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.) (New usage is discouraged.)

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	. . . . 5 ⊢ Ⅎ𝑧𝜑
2	1	nfn 1851	. . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑
3		cbval2.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	nfn 1851	. . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑
5		cbval2.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1851	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		cbval2.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	7	nfn 1851	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓
9		cbval2.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
10	9	notbid 320	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
11	2, 4, 6, 8, 10	cbval2 2426	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
12		2nexaln 1824	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
13		2nexaln 1824	. . 3 ⊢ (¬ ∃𝑧∃𝑤𝜓 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
14	11, 12, 13	3bitr4i 305	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ¬ ∃𝑧∃𝑤𝜓)
15	14	con4bii 323	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1529  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779

This theorem is referenced by: (None)