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Theorem vtoclri 3573
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
Hypotheses
Ref Expression
vtoclri.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclri.2 𝑥𝐵 𝜑
Assertion
Ref Expression
vtoclri (𝐴𝐵𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclri
StepHypRef Expression
1 vtoclri.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
2 vtoclri.2 . . 3 𝑥𝐵 𝜑
32rspec 3246 . 2 (𝑥𝐵𝜑)
41, 3vtoclga 3562 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061
This theorem is referenced by:  alephreg  10559  arch  12451  harmonicbnd  26435  harmonicbnd2  26436  nbgrnself2  28482  heiborlem8  36491  fourierdlem62  44657  natglobalincr  45364  srhmsubclem1  46619  srhmsubc  46622  srhmsubcALTV  46640
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