ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbceq2a Unicode version

Theorem sbceq2a 2973
Description: Equality theorem for class substitution. Class version of sbequ12r 1772. (Contributed by NM, 4-Jan-2017.)
Assertion
Ref Expression
sbceq2a  |-  ( A  =  x  ->  ( [. A  /  x ]. ph  <->  ph ) )

Proof of Theorem sbceq2a
StepHypRef Expression
1 sbceq1a 2972 . . 3  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21eqcoms 2180 . 2  |-  ( A  =  x  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
32bicomd 141 1  |-  ( A  =  x  ->  ( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator