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Theorem abbid 2204
Description: Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
abbid.1  |-  F/ x ph
abbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
abbid  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )

Proof of Theorem abbid
StepHypRef Expression
1 abbid.1 . . 3  |-  F/ x ph
2 abbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2alrimi 1460 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 abbi 2201 . 2  |-  ( A. x ( ps  <->  ch )  <->  { x  |  ps }  =  { x  |  ch } )
53, 4sylib 120 1  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394   {cab 2074
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
This theorem is referenced by:  abbidv  2205  rabeqf  2609  sbcbid  2894
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