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Theorem cbvexd 2266
 Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2325. (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 1769 . . . 4 (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4 cbvald.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
5 notbi 308 . . . . 5 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
64, 5syl6ib 240 . . . 4 (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
71, 3, 6cbvald 2265 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
87notbid 307 . 2 (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒))
9 df-ex 1696 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
10 df-ex 1696 . 2 (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
118, 9, 103bitr4g 302 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 Colors of variables: wff setvar class 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  ax-13 2234 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:  cbvexdvaOLD  2272  vtoclgft  3227  vtoclgftOLD  3228  dfid3  4954  axrepndlem2  9294  axunnd  9297  axpowndlem2  9299  axpownd  9302  axregndlem2  9304  axinfndlem1  9306  axacndlem4  9311  wl-mo2df  32531  wl-eudf  32533  wl-mo2t  32536
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