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Theorem cbvexd 2372
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2416. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbvald.1 𝑦𝜑
cbvald.2 (𝜑 → Ⅎ𝑦𝜓)
cbvald.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvexd (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvald.1 . . . 4 𝑦𝜑
2 cbvald.2 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
32nfnd 1903 . . . 4 (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4 cbvald.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
5 notbi 311 . . . . 5 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
64, 5syl6ib 243 . . . 4 (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
71, 3, 6cbvald 2371 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8 alnex 1825 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9 alnex 1825 . . 3 (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
107, 8, 93bitr3g 305 . 2 (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
1110con4bid 309 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1599  wex 1823  wnf 1827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828
This theorem is referenced by:  vtoclgft  3455  dfid3  5262  axrepndlem2  9750  axunnd  9753  axpowndlem2  9755  axpownd  9758  axregndlem2  9760  axinfndlem1  9762  axacndlem4  9767  wl-mo2df  33960  wl-eudf  33962  wl-mo2t  33965
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