MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrab Unicode version

Theorem nfrab 2721
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 2552 . 2  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
2 nftru 1541 . . . 4  |-  F/ y  T.
3 nfrab.2 . . . . . . . 8  |-  F/_ x A
43nfcri 2413 . . . . . . 7  |-  F/ x  z  e.  A
5 eleq1 2343 . . . . . . 7  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
64, 5dvelimnf 1957 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x  y  e.  A )
7 nfrab.1 . . . . . . 7  |-  F/ x ph
87a1i 10 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  F/ x ph )
96, 8nfand 1763 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  F/ x
( y  e.  A  /\  ph ) )
109adantl 452 . . . 4  |-  ( (  T.  /\  -.  A. x  x  =  y
)  ->  F/ x
( y  e.  A  /\  ph ) )
112, 10nfabd2 2437 . . 3  |-  (  T. 
->  F/_ x { y  |  ( y  e.  A  /\  ph ) } )
1211trud 1314 . 2  |-  F/_ x { y  |  ( y  e.  A  /\  ph ) }
131, 12nfcxfr 2416 1  |-  F/_ x { y  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    T. wtru 1307   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   {crab 2547
This theorem is referenced by:  nfdif  3297  nfin  3375  nfse  4368  reusv6OLD  4545  nfoi  7229  scottex  7555  elmptrab  17522  iundisjf  23365  finminlem  26231  indexa  26412  stoweidlem16  27765  stoweidlem31  27780  stoweidlem34  27783  stoweidlem35  27784  stoweidlem48  27797  stoweidlem51  27800  stoweidlem53  27802  stoweidlem54  27803  stoweidlem57  27806  stoweidlem59  27808  mpt2xopoveq  28085  bnj1398  29064  bnj1445  29074  bnj1449  29078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552
  Copyright terms: Public domain W3C validator