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Theorem rabbiia 2722
Description: Equivalent wff's yield equal restricted class abstractions (inference form). (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 454 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32abbii 2293 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  ps ) }
4 df-rab 2464 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
5 df-rab 2464 . 2  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
63, 4, 53eqtr4i 2208 1  |-  { x  e.  A  |  ph }  =  { x  e.  A  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   {crab 2459
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  df-rab 2464
This theorem is referenced by:  rabbii  2723  bm2.5ii  4495  fndmdifcom  5622  cauappcvgprlemladdru  7654  cauappcvgprlemladdrl  7655  cauappcvgpr  7660  caucvgprlemcl  7674  caucvgprlemladdrl  7676  caucvgpr  7680  caucvgprprlemclphr  7703  ioopos  9948  gcdcom  11968  gcdass  12010  lcmcom  12058  lcmass  12079
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