ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclef Unicode version

Theorem vtoclef 2731
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
vtoclef.1  |-  F/ x ph
vtoclef.2  |-  A  e. 
_V
vtoclef.3  |-  ( x  =  A  ->  ph )
Assertion
Ref Expression
vtoclef  |-  ph
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem vtoclef
StepHypRef Expression
1 vtoclef.2 . . 3  |-  A  e. 
_V
21isseti 2666 . 2  |-  E. x  x  =  A
3 vtoclef.1 . . 3  |-  F/ x ph
4 vtoclef.3 . . 3  |-  ( x  =  A  ->  ph )
53, 4exlimi 1556 . 2  |-  ( E. x  x  =  A  ->  ph )
62, 5ax-mp 5 1  |-  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463   _Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  nn0ind-raph  9122
  Copyright terms: Public domain W3C validator