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Theorem sbcbid 2894
Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcbid.1  |-  F/ x ph
sbcbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
sbcbid  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4  |-  F/ x ph
2 sbcbid.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2abbid 2204 . . 3  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
43eleq2d 2157 . 2  |-  ( ph  ->  ( A  e.  {
x  |  ps }  <->  A  e.  { x  |  ch } ) )
5 df-sbc 2839 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
6 df-sbc 2839 . 2  |-  ( [. A  /  x ]. ch  <->  A  e.  { x  |  ch } )
74, 5, 63bitr4g 221 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394    e. wcel 1438   {cab 2074   [.wsbc 2838
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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839
This theorem is referenced by:  sbcbidv  2895  csbeq2d  2953  bezoutlemstep  11079
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