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Theorem bj-cbvexdv 32431
Description: Version of cbvexd 2277 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbvaldv.1 𝑦𝜑
bj-cbvaldv.2 (𝜑 → Ⅎ𝑦𝜓)
bj-cbvaldv.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
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
StepHypRef Expression
1 bj-cbvaldv.1 . . . 4 𝑦𝜑
2 bj-cbvaldv.2 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
32nfnd 1782 . . . 4 (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4 bj-cbvaldv.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
5 notbi 309 . . . . 5 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
64, 5syl6ib 241 . . . 4 (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
71, 3, 6bj-cbvaldv 32430 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
87notbid 308 . 2 (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒))
9 df-ex 1702 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
10 df-ex 1702 . 2 (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
118, 9, 103bitr4g 303 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-cbvexdvav  32437
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