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Theorem vtoclri 3256
 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 2915 . 2 (𝑥𝐵𝜑)
41, 3vtoclga 3245 1 (𝐴𝐵𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175 This theorem is referenced by:  alephreg  9283  arch  11166  harmonicbnd  24530  harmonicbnd2  24531  heiborlem8  32787  fourierdlem62  39061  srhmsubclem1  41865  srhmsubc  41868  srhmsubcALTVlem1  41884  srhmsubcALTV  41887
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