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

Theorem vtoclef 3272
 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 3200 . 2 𝑥 𝑥 = 𝐴
3 vtoclef.1 . . 3 𝑥𝜑
4 vtoclef.3 . . 3 (𝑥 = 𝐴𝜑)
53, 4exlimi 2089 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
62, 5ax-mp 5 1 𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480  ∃wex 1701  Ⅎwnf 1705   ∈ wcel 1992  Vcvv 3191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-12 2049  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-v 3193 This theorem is referenced by:  nn0ind-raph  11421  finxpreclem2  32832
 Copyright terms: Public domain W3C validator