MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocle Structured version   Unicode version

Theorem vtocle 3187
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
Hypotheses
Ref Expression
vtocle.1  |-  A  e. 
_V
vtocle.2  |-  ( x  =  A  ->  ph )
Assertion
Ref Expression
vtocle  |-  ph
Distinct variable groups:    x, A    ph, x

Proof of Theorem vtocle
StepHypRef Expression
1 vtocle.1 . 2  |-  A  e. 
_V
2 vtocle.2 . . 3  |-  ( x  =  A  ->  ph )
32vtocleg 3184 . 2  |-  ( A  e.  _V  ->  ph )
41, 3ax-mp 5 1  |-  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  zfrepclf  4564  tz6.12i  5884  eloprabga  6371  cfflb  8635  axcc3  8814  nn0ind-raph  10957
  Copyright terms: Public domain W3C validator