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Theorem cbvald 2428
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2473. Usage of this theorem is discouraged because it depends on ax-13 2390. See cbvaldw 2358 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2390. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvald.1 𝑦𝜑
cbvald.2 (𝜑 → Ⅎ𝑦𝜓)
cbvald.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvald (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvald
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜑
2 cbvald.1 . 2 𝑦𝜑
3 cbvald.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfvd 1916 . 2 (𝜑 → Ⅎ𝑥𝜒)
5 cbvald.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 3, 4, 5cbv2 2423 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1535  wnf 1784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-12 2177  ax-13 2390
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785
This theorem is referenced by:  cbvexd  2429  cbvaldva  2430  axextnd  10015  axrepndlem1  10016  axunndlem1  10019  axpowndlem2  10022  axpowndlem3  10023  axpowndlem4  10024  axregndlem2  10027  axregnd  10028  axinfnd  10030  axacndlem5  10035  axacnd  10036  axextdist  33046  distel  33050  wl-sb8eut  34815
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