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Theorem rabid2f 23150
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rabid2f.1 . . . . . 6  |-  F/_ x A
21nfcrii 2487 . . . . 5  |-  ( y  e.  A  ->  A. x  y  e.  A )
3 hbab1 2347 . . . . 5  |-  ( y  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  A. x  y  e.  {
x  |  ( x  e.  A  /\  ph ) } )
42, 3cleqh 2455 . . . 4  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  x  e.  { x  |  ( x  e.  A  /\  ph ) } ) )
5 abid 2346 . . . . . 6  |-  ( x  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  ( x  e.  A  /\  ph )
)
65bibi2i 304 . . . . 5  |-  ( ( x  e.  A  <->  x  e.  { x  |  ( x  e.  A  /\  ph ) } )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
76albii 1566 . . . 4  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ( x  e.  A  /\  ph ) } )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
84, 7bitri 240 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
9 pm4.71 611 . . . 4  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  ph )
) )
109albii 1566 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  A  <->  ( x  e.  A  /\  ph ) ) )
118, 10bitr4i 243 . 2  |-  ( A  =  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  A  ->  ph ) )
12 df-rab 2628 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
1312eqeq2i 2368 . 2  |-  ( A  =  { x  e.  A  |  ph }  <->  A  =  { x  |  ( x  e.  A  /\  ph ) } )
14 df-ral 2624 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
1511, 13, 143bitr4i 268 1  |-  ( A  =  { x  e.  A  |  ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710   {cab 2344   F/_wnfc 2481   A.wral 2619   {crab 2623
This theorem is referenced by:  funcnvmptOLD  23282  funcnvmpt  23283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628
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