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Theorem vtoclri 3534
 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 3172 . 2 (𝑥𝐵𝜑)
41, 3vtoclga 3523 1 (𝐴𝐵𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111 This theorem is referenced by:  alephreg  10011  arch  11900  harmonicbnd  25633  harmonicbnd2  25634  nbgrnself2  27194  heiborlem8  35407  fourierdlem62  42978  srhmsubclem1  44865  srhmsubc  44868  srhmsubcALTV  44886
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