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Theorem rabbiia 2596
Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
Hypothesis
Ref Expression
rabbiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rabbiia  |-  { x  e.  A  |  ph }  =  { x  e.  A  |  ps }

Proof of Theorem rabbiia
StepHypRef Expression
1 rabbiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 442 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32abbii 2198 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  ps ) }
4 df-rab 2362 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
5 df-rab 2362 . 2  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
63, 4, 53eqtr4i 2113 1  |-  { x  e.  A  |  ph }  =  { x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   {crab 2357
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-rab 2362
This theorem is referenced by:  rabbii  2597  bm2.5ii  4268  fndmdifcom  5325  cauappcvgprlemladdru  6960  cauappcvgprlemladdrl  6961  cauappcvgpr  6966  caucvgprlemcl  6980  caucvgprlemladdrl  6982  caucvgpr  6986  caucvgprprlemclphr  7009  ioopos  9101  gcdcom  10572  gcdass  10611  lcmcom  10653  lcmass  10674
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