Theorem bj-cbvexdv 31923

Description: Version of cbvexd 2266 with a dv condition, which does not require ax-13 2234. (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 1769	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4		bj-cbvaldv.3	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5		notbi 308	. . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
6	4, 5	syl6ib 240	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
7	1, 3, 6	bj-cbvaldv 31922	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8	7	notbid 307	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒))
9		df-ex 1696	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
10		df-ex 1696	. 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
11	8, 9, 10	3bitr4g 302	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-cbvexdvav  31929