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Theorem cbvald 2427
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2472. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
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 2013 . 2 𝑥𝜑
2 cbvald.1 . 2 𝑦𝜑
3 cbvald.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfvd 2014 . 2 (𝜑 → Ⅎ𝑥𝜒)
5 cbvald.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 3, 4, 5cbv2 2422 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1654  wnf 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883
This theorem is referenced by:  cbvexd  2428  axextnd  9735  axrepndlem1  9736  axunndlem1  9739  axpowndlem2  9742  axpowndlem3  9743  axpowndlem4  9744  axregndlem2  9747  axregnd  9748  axinfnd  9750  axacndlem5  9755  axacnd  9756  axextdist  32238  distel  32242  wl-sb8eut  33898
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