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Theorem vtoclef 3143
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
vtoclef.1 𝑥𝜑
vtoclef.2 𝐴 ∈ V
vtoclef.3 (𝑥 = 𝐴𝜑)
Assertion
Ref Expression
vtoclef 𝜑
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclef
StepHypRef Expression
1 vtoclef.2 . . 3 𝐴 ∈ V
21isseti 3072 . 2 𝑥 𝑥 = 𝐴
3 vtoclef.1 . . 3 𝑥𝜑
4 vtoclef.3 . . 3 (𝑥 = 𝐴𝜑)
53, 4exlimi 2048 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
62, 5ax-mp 5 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1468  wex 1692  wnf 1696  wcel 1937  Vcvv 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-v 3068
This theorem is referenced by:  nn0ind-raph  11125  finxpreclem2  32003
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