Theorem bj-cbvex2v 32738

Description: Version of cbvex2 2280 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbval2v.1	⊢ Ⅎ𝑧𝜑
bj-cbval2v.2	⊢ Ⅎ𝑤𝜑
bj-cbval2v.3	⊢ Ⅎ𝑥𝜓
bj-cbval2v.4	⊢ Ⅎ𝑦𝜓
bj-cbval2v.5	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-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 bj-cbvex2v

Step	Hyp	Ref	Expression
1		bj-cbval2v.1	. . . . 5 ⊢ Ⅎ𝑧𝜑
2	1	nfn 1784	. . . 4 ⊢ Ⅎ𝑧 ¬ 𝜑
3		bj-cbval2v.2	. . . . 5 ⊢ Ⅎ𝑤𝜑
4	3	nfn 1784	. . . 4 ⊢ Ⅎ𝑤 ¬ 𝜑
5		bj-cbval2v.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1784	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
7		bj-cbval2v.4	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	7	nfn 1784	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜓
9		bj-cbval2v.5	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
10	9	notbid 308	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
11	2, 4, 6, 8, 10	bj-cbval2v 32737	. . 3 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑧∀𝑤 ¬ 𝜓)
12	11	notbii 310	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓)
13		2exnaln 1756	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
14		2exnaln 1756	. 2 ⊢ (∃𝑧∃𝑤𝜓 ↔ ¬ ∀𝑧∀𝑤 ¬ 𝜓)
15	12, 13, 14	3bitr4i 292	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-cbvex2vv  32740