Theorem bj-cbvexdv 37154

Description: Version of cbvexd 2416 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbvaldv.1	⊢ Ⅎ𝑦𝜑
bj-cbvaldv.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
bj-cbvaldv.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
bj-cbvexdv	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem bj-cbvexdv

Step	Hyp	Ref	Expression
1		bj-cbvaldv.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		bj-cbvaldv.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3	2	nfnd 1865	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4		bj-cbvaldv.3	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5		notbi 320	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
6	4, 5	imbitrdi 252	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
7	1, 3, 6	bj-cbvaldv 37153	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8	7	notbid 319	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒))
9		df-ex 1787	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
10		df-ex 1787	. 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
11	8, 9, 10	3bitr4g 315	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  bj-cbvexdvav  37158