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Theorem sbcbid 3159
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 2502 . . 3  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
43eleq2d 2456 . 2  |-  ( ph  ->  ( A  e.  {
x  |  ps }  <->  A  e.  { x  |  ch } ) )
5 df-sbc 3107 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
6 df-sbc 3107 . 2  |-  ( [. A  /  x ]. ch  <->  A  e.  { x  |  ch } )
74, 5, 63bitr4g 280 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550    e. wcel 1717   {cab 2375   [.wsbc 3106
This theorem is referenced by:  sbcbidv  3160  csbeq2d  3220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-sbc 3107
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