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Theorem abbid 2396
Description: Equivalent wff's yield equal class abstractions (deduction rule). (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 1745 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 abbi 2393 . 2  |-  ( A. x ( ps  <->  ch )  <->  { x  |  ps }  =  { x  |  ch } )
53, 4sylib 188 1  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623   {cab 2269
This theorem is referenced by:  abbidv  2397  rabeqf  2781  sbcbid  3044  iotain  27617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276
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