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Theorem cbvexdw 2361
Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2431 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvaldw.1 𝑦𝜑
cbvaldw.2 (𝜑 → Ⅎ𝑦𝜓)
cbvaldw.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvexdw (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvexdw
StepHypRef Expression
1 cbvaldw.1 . . . 4 𝑦𝜑
2 cbvaldw.2 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
32nfnd 1859 . . . 4 (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4 cbvaldw.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
5 notbi 322 . . . . 5 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
64, 5syl6ib 254 . . . 4 (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
71, 3, 6cbvaldw 2360 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8 alnex 1783 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9 alnex 1783 . . 3 (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
107, 8, 93bitr3g 316 . 2 (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
1110con4bid 320 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  bj-opabco  34575  wl-mo2t  34948
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