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Theorem vtoclri 3504
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 3157 . 2 (𝑥𝐵𝜑)
41, 3vtoclga 3493 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wcel 2050  wral 3088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-cleq 2771  df-clel 2846  df-ral 3093
This theorem is referenced by:  alephreg  9802  arch  11704  harmonicbnd  25283  harmonicbnd2  25284  nbgrnself2  26845  heiborlem8  34544  fourierdlem62  41890  srhmsubclem1  43714  srhmsubc  43717  srhmsubcALTV  43735
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