MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfriota1 Structured version   Unicode version

Theorem nfriota1 6557
Description: The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
nfriota1  |-  F/_ x
( iota_ x  e.  A ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem nfriota1
StepHypRef Expression
1 df-riota 6549 . 2  |-  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
2 nfreu1 2878 . . 3  |-  F/ x E! x  e.  A  ph
3 nfiota1 5420 . . 3  |-  F/_ x
( iota x ( x  e.  A  /\  ph ) )
4 nfcv 2572 . . . 4  |-  F/_ x Undef
5 nfab1 2574 . . . 4  |-  F/_ x { x  |  x  e.  A }
64, 5nffv 5735 . . 3  |-  F/_ x
( Undef `  { x  |  x  e.  A } )
72, 3, 6nfif 3763 . 2  |-  F/_ x if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
81, 7nfcxfr 2569 1  |-  F/_ x
( iota_ x  e.  A ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725   {cab 2422   F/_wnfc 2559   E!wreu 2707   ifcif 3739   iotacio 5416   ` cfv 5454   Undefcund 6541   iota_crio 6542
This theorem is referenced by:  riotaprop  6573  riotass2  6577  riotass  6578  riotaxfrd  6581  riotasvdOLD  6593  lble  9960  riotaneg  9983  riotaocN  30007  ltrniotaval  31378  cdlemksv2  31644  cdlemkuv2  31664  cdlemk36  31710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-riota 6549
  Copyright terms: Public domain W3C validator