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

Theorem sbcieg 2969
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
Hypothesis
Ref Expression
sbcieg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbcieg  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem sbcieg
StepHypRef Expression
1 nfv 1508 . 2  |-  F/ x ps
2 sbcieg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2sbciegf 2968 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128   [.wsbc 2937
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938
This theorem is referenced by:  sbcie  2971  ralsng  3600  rexsng  3601  ralrnmpt  5610  rexrnmpt  5611  nn1suc  8853  cjth  10750  bezoutlemnewy  11884  bezoutlemstep  11885  bezoutlema  11887  bezoutlemb  11888  prmind2  12001
  Copyright terms: Public domain W3C validator