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Theorem vtoclri 3501
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 3129 . 2 (𝑥𝐵𝜑)
41, 3vtoclga 3489 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066
This theorem is referenced by:  alephreg  10196  arch  12087  harmonicbnd  25886  harmonicbnd2  25887  nbgrnself2  27448  heiborlem8  35713  fourierdlem62  43384  srhmsubclem1  45304  srhmsubc  45307  srhmsubcALTV  45325
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