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Theorem rabid2f 23920
Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
rabid2f.1  |-  F/_ x A
Assertion
Ref Expression
rabid2f  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )

Proof of Theorem rabid2f
StepHypRef Expression
1 rabid2f.1 . . . 4  |-  F/_ x A
21abeq2f 23913 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
3 pm4.71 612 . . . 4  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
43albii 1572 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
52, 4bitr4i 244 . 2  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  ->  ph ) )
6 df-rab 2675 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
76eqeq2i 2414 . 2  |-  ( A  =  { x  e.  A  |  ph }  <->  A  =  { x  |  ( x  e.  A  /\  ph ) } )
8 df-ral 2671 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
95, 7, 83bitr4i 269 1  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2390   F/_wnfc 2527   A.wral 2666   {crab 2670
This theorem is referenced by:  funcnvmptOLD  24035  funcnvmpt  24036
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  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675
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