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Theorem nfrab 2832
Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
nfrab.1  |-  F/ x ph
nfrab.2  |-  F/_ x A
Assertion
Ref Expression
nfrab  |-  F/_ x { y  e.  A  |  ph }

Proof of Theorem nfrab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-rab 2658 . 2  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
2 nftru 1560 . . . 4  |-  F/ y  T.
3 nfrab.2 . . . . . . . 8  |-  F/_ x A
43nfcri 2517 . . . . . . 7  |-  F/ x  z  e.  A
5 eleq1 2447 . . . . . . 7  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
64, 5dvelimnf 2051 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  e.  A )
7 nfrab.1 . . . . . . 7  |-  F/ x ph
87a1i 11 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
96, 8nfand 1833 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x
( y  e.  A  /\  ph ) )
109adantl 453 . . . 4  |-  ( (  T.  /\  -.  A. x  x  =  y
)  ->  F/ x
( y  e.  A  /\  ph ) )
112, 10nfabd2 2541 . . 3  |-  (  T. 
->  F/_ x { y  |  ( y  e.  A  /\  ph ) } )
1211trud 1329 . 2  |-  F/_ x { y  |  ( y  e.  A  /\  ph ) }
131, 12nfcxfr 2520 1  |-  F/_ x { y  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    T. wtru 1322   A.wal 1546   F/wnf 1550    e. wcel 1717   {cab 2373   F/_wnfc 2510   {crab 2653
This theorem is referenced by:  nfdif  3411  nfin  3490  nfse  4498  reusv6OLD  4674  mpt2xopoveq  6406  nfoi  7416  scottex  7742  elmptrab  17780  iundisjf  23872  finminlem  26012  indexa  26126  stoweidlem16  27433  stoweidlem31  27448  stoweidlem34  27451  stoweidlem35  27452  stoweidlem48  27465  stoweidlem51  27468  stoweidlem53  27470  stoweidlem54  27471  stoweidlem57  27474  stoweidlem59  27476  bnj1398  28741  bnj1445  28751  bnj1449  28755
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658
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