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

Theorem vtocleg 3142
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
vtocleg.1  |-  ( x  =  A  ->  ph )
Assertion
Ref Expression
vtocleg  |-  ( A  e.  V  ->  ph )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    V( x)

Proof of Theorem vtocleg
StepHypRef Expression
1 elisset 3082 . 2  |-  ( A  e.  V  ->  E. x  x  =  A )
2 vtocleg.1 . . 3  |-  ( x  =  A  ->  ph )
32exlimiv 1689 . 2  |-  ( E. x  x  =  A  ->  ph )
41, 3syl 16 1  |-  ( A  e.  V  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   E.wex 1587    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3073
This theorem is referenced by:  vtocle  3145  spsbc  3300  prex  4635  avril1  23801
  Copyright terms: Public domain W3C validator