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Theorem sbceq1d 2845
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 2842 . 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 1289   [.wsbc 2840
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-sbc 2841
This theorem is referenced by:  sbceq1dd  2846  rexrnmpt  5442  findcard2  6605  findcard2s  6606  ac6sfi  6614  nn1suc  8441  uzind4s  9078  uzind4s2  9079  fzrevral  9519  fzshftral  9522  cjth  10280  prmind2  11380  bj-bdfindes  11844  bj-findes  11876
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