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Theorem sbcbid 2843
Description: Formula-building deduction rule 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 2170 . . 3  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
43eleq2d 2123 . 2  |-  ( ph  ->  ( A  e.  {
x  |  ps }  <->  A  e.  { x  |  ch } ) )
5 df-sbc 2788 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
6 df-sbc 2788 . 2  |-  ( [. A  /  x ]. ch  <->  A  e.  { x  |  ch } )
74, 5, 63bitr4g 216 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   F/wnf 1365    e. wcel 1409   {cab 2042   [.wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-sbc 2788
This theorem is referenced by:  sbcbidv  2844  csbeq2d  2902
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