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Theorem vtoclf 3231
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2250. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtoclf.1 𝑥𝜓
vtoclf.2 𝐴 ∈ V
vtoclf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclf.4 𝜑
Assertion
Ref Expression
vtoclf 𝜓
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclf
StepHypRef Expression
1 vtoclf.1 . . 3 𝑥𝜓
2 vtoclf.2 . . . . 5 𝐴 ∈ V
32isseti 3182 . . . 4 𝑥 𝑥 = 𝐴
4 vtoclf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 218 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
63, 5eximii 1754 . . 3 𝑥(𝜑𝜓)
71, 619.36i 2086 . 2 (∀𝑥𝜑𝜓)
8 vtoclf.4 . 2 𝜑
97, 8mpg 1715 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wnf 1699  wcel 1977  Vcvv 3173
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-12 2034  ax-ext 2590
This theorem depends on definitions:  df-bi 196  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-v 3175
This theorem is referenced by:  vtoclALT  3233  summolem2a  14242  prodmolem2a  14452  poimirlem24  32397  poimirlem28  32401  monotuz  36318  oddcomabszz  36321  binomcxplemnotnn0  37371  limclner  38512  dvnmptdivc  38622  dvnmul  38627  salpreimagtge  39405  salpreimaltle  39406
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