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Theorem sbceq1d 2821
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 2818 . 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 103    = wceq 1285   [.wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-sbc 2817
This theorem is referenced by:  sbceq1dd  2822  rexrnmpt  5342  findcard2  6423  findcard2s  6424  ac6sfi  6431  nn1suc  8125  uzind4s  8759  uzind4s2  8760  fzrevral  9198  fzshftral  9201  cjth  9871  prmind2  10646  bj-bdfindes  10902  bj-findes  10934
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