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

Theorem vtoclef 3499
 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 3423 . 2 𝑥 𝑥 = 𝐴
3 vtoclef.1 . . 3 𝑥𝜑
4 vtoclef.3 . . 3 (𝑥 = 𝐴𝜑)
53, 4exlimi 2216 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
62, 5ax-mp 5 1 𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2112  Vcvv 3407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-cleq 2751  df-clel 2831 This theorem is referenced by:  nn0ind-raph  12106  rdgssun  35060  finxpreclem2  35072
 Copyright terms: Public domain W3C validator