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Theorem sbcbid 3020
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 2294 . . 3  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
43eleq2d 2247 . 2  |-  ( ph  ->  ( A  e.  {
x  |  ps }  <->  A  e.  { x  |  ch } ) )
5 df-sbc 2963 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
6 df-sbc 2963 . 2  |-  ( [. A  /  x ]. ch  <->  A  e.  { x  |  ch } )
74, 5, 63bitr4g 223 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1460    e. wcel 2148   {cab 2163   [.wsbc 2962
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2963
This theorem is referenced by:  sbcbidv  3021  csbeq2d  3082  bezoutlemstep  11997
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