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Theorem cbvald 2410
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2454. Usage of this theorem is discouraged because it depends on ax-13 2375. See cbvaldw 2339 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2375. (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 1912 . 2 𝑥𝜑
2 cbvald.1 . 2 𝑦𝜑
3 cbvald.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfvd 1913 . 2 (𝜑 → Ⅎ𝑥𝜒)
5 cbvald.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 3, 4, 5cbv2 2406 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  cbvexd  2411  cbvaldva  2412  axextnd  10629  axrepndlem1  10630  axunndlem1  10633  axpowndlem2  10636  axpowndlem3  10637  axpowndlem4  10638  axregndlem2  10641  axregnd  10642  axinfnd  10644  axacndlem5  10649  axacnd  10650  axextdist  35781  distel  35785  wl-sb8eut  37559
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