Theorem cbvexv1 2164

Description: Version of cbvex 2260 with a dv condition, which does not require ax-13 2234. See cbvexvw 1957 for a version with two dv conditions, requiring fewer axioms, and cbvexv 2263 for another variant. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	⊢ Ⅎ𝑦𝜑
cbvalv1.nf2	⊢ Ⅎ𝑥𝜓
cbvalv1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvexv1	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1768	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
3		cbvalv1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	3	nfn 1768	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5		cbvalv1.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	notbid 307	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
7	2, 4, 6	cbvalv1 2163	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
8	7	notbii 309	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓)
9		df-ex 1696	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
10		df-ex 1696	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
11	8, 9, 10	3bitr4i 291	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

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-cbvexvv  31921  bj-axrep1  31976  bj-axrep2  31977  bj-axrep4  31979