MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbceq1dd Structured version   Unicode version

Theorem sbceq1dd 3168
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 3167 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ps ) )
41, 3mpbid 203 1  |-  ( ph  ->  [. B  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   [.wsbc 3162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2430  df-clel 2433  df-sbc 3163
  Copyright terms: Public domain W3C validator