MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abbid Unicode version

Theorem abbid 2409
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 1757 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 abbi 2406 . 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 1530   F/wnf 1534    = wceq 1632   {cab 2282
This theorem is referenced by:  abbidv  2410  rabeqf  2794  sbcbid  3057  iotain  27720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289
  Copyright terms: Public domain W3C validator