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Theorem abbid 2517
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 1777 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 abbi 2514 . 2  |-  ( A. x ( ps  <->  ch )  <->  { x  |  ps }  =  { x  |  ch } )
53, 4sylib 189 1  |-  ( ph  ->  { x  |  ps }  =  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   F/wnf 1550    = wceq 1649   {cab 2390
This theorem is referenced by:  abbidv  2518  rabeqf  2909  sbcbid  3174  iotain  27485
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397
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