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Theorem sbceq1d 2982
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 2979 . 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 105    = wceq 1364   [.wsbc 2977
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-sbc 2978
This theorem is referenced by:  sbceq1dd  2983  rexrnmpt  5680  findcard2  6917  findcard2s  6918  ac6sfi  6926  nn1suc  8968  uzind4s  9620  uzind4s2  9621  fzrevral  10135  fzshftral  10138  cjth  10887  prmind2  12152  issrg  13319  islmod  13607  bj-bdfindes  15162  bj-findes  15194
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