MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtoclef Structured version   Visualization version   GIF version

Theorem vtoclef 3588
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 3514 . 2 𝑥 𝑥 = 𝐴
3 vtoclef.1 . . 3 𝑥𝜑
4 vtoclef.3 . . 3 (𝑥 = 𝐴𝜑)
53, 4exlimi 2210 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
62, 5ax-mp 5 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wex 1773  wnf 1777  wcel 2107  Vcvv 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-cleq 2819  df-clel 2898
This theorem is referenced by:  nn0ind-raph  12071  rdgssun  34528  finxpreclem2  34540
  Copyright terms: Public domain W3C validator