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Theorem cbvexv1 2212
Description: Version of cbvex 2308 with a dv condition, which does not require ax-13 2282. See cbvexvw 2012 for a version with two dv conditions, requiring fewer axioms, and cbvexv 2311 for another variant. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
cbvalv1.nf1 𝑦𝜑
cbvalv1.nf2 𝑥𝜓
cbvalv1.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexv1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvexv1
StepHypRef Expression
1 cbvalv1.nf1 . . . . 5 𝑦𝜑
21nfn 1824 . . . 4 𝑦 ¬ 𝜑
3 cbvalv1.nf2 . . . . 5 𝑥𝜓
43nfn 1824 . . . 4 𝑥 ¬ 𝜓
5 cbvalv1.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
65notbid 307 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
72, 4, 6cbvalv1 2211 . . 3 (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓)
87notbii 309 . 2 (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓)
9 df-ex 1745 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
10 df-ex 1745 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
118, 9, 103bitr4i 292 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  bj-cbvexvv  32859  bj-axrep1  32913  bj-axrep2  32914  bj-axrep4  32916
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