ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbceq1d Unicode version

Theorem sbceq1d 2887
Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
Hypothesis
Ref Expression
sbceq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
sbceq1d  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ps ) )

Proof of Theorem sbceq1d
StepHypRef Expression
1 sbceq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 dfsbcq 2884 . 2  |-  ( A  =  B  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ps ) )
31, 2syl 14 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   [.wsbc 2882
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113  df-sbc 2883
This theorem is referenced by:  sbceq1dd  2888  rexrnmpt  5531  findcard2  6751  findcard2s  6752  ac6sfi  6760  nn1suc  8703  uzind4s  9341  uzind4s2  9342  fzrevral  9840  fzshftral  9843  cjth  10573  prmind2  11713  bj-bdfindes  13043  bj-findes  13075
  Copyright terms: Public domain W3C validator