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

Theorem nfriota1 6328
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 6320 . 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 2723 . . 3  |-  F/ x E! x  e.  A  ph
3 nfiota1 5237 . . 3  |-  F/_ x
( iota x ( x  e.  A  /\  ph ) )
4 nfcv 2432 . . . 4  |-  F/_ x Undef
5 nfab1 2434 . . . 4  |-  F/_ x { x  |  x  e.  A }
64, 5nffv 5548 . . 3  |-  F/_ x
( Undef `  { x  |  x  e.  A } )
72, 3, 6nfif 3602 . 2  |-  F/_ x if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
81, 7nfcxfr 2429 1  |-  F/_ x
( iota_ x  e.  A ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696   {cab 2282   F/_wnfc 2419   E!wreu 2558   ifcif 3578   iotacio 5233   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem is referenced by:  riotaprop  6344  riotass2  6348  riotass  6349  riotaxfrd  6352  riotasvdOLD  6364  lble  9722  riotaneg  9745  lineval22  26185  riotaocN  30021  ltrniotaval  31392  cdlemksv2  31658  cdlemkuv2  31678  cdlemk36  31724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
  Copyright terms: Public domain W3C validator