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Theorem sbceq1dd 2961
Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
Hypotheses
Ref Expression
sbceq1d.1  |-  ( ph  ->  A  =  B )
sbceq1dd.2  |-  ( ph  ->  [. A  /  x ]. ps )
Assertion
Ref Expression
sbceq1dd  |-  ( ph  ->  [. B  /  x ]. ps )

Proof of Theorem sbceq1dd
StepHypRef Expression
1 sbceq1dd.2 . 2  |-  ( ph  ->  [. A  /  x ]. ps )
2 sbceq1d.1 . . 3  |-  ( ph  ->  A  =  B )
32sbceq1d 2960 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ps ) )
41, 3mpbid 146 1  |-  ( ph  ->  [. B  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   [.wsbc 2955
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-sbc 2956
This theorem is referenced by:  prmind2  12074
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