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Theorem abbid 2294
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 1522 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 abbi 2291 . 2  |-  ( A. x ( ps  <->  ch )  <->  { x  |  ps }  =  { x  |  ch } )
53, 4sylib 122 1  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353   F/wnf 1460   {cab 2163
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
This theorem is referenced by:  abbidv  2295  rabeqf  2727  sbcbid  3020
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