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Theorem sbceqbid 3190
Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
Hypotheses
Ref Expression
sbceqbid.1  |-  ( ph  ->  A  =  B )
sbceqbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
sbceqbid  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem sbceqbid
StepHypRef Expression
1 sbceqbid.1 . . 3  |-  ( ph  ->  A  =  B )
2 sbceqbid.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
32abbidv 2555 . . 3  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
41, 3eleq12d 2509 . 2  |-  ( ph  ->  ( A  e.  {
x  |  ps }  <->  B  e.  { x  |  ch } ) )
5 df-sbc 3184 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
6 df-sbc 3184 . 2  |-  ( [. B  /  x ]. ch  <->  B  e.  { x  |  ch } )
74, 5, 63bitr4g 288 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1364    e. wcel 1761   {cab 2427   [.wsbc 3183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-sbc 3184
This theorem is referenced by:  issrg  16599
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