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Theorem riotaiota 6555
Description: Restricted iota in terms of iota. (Contributed by NM, 15-Sep-2011.)
Assertion
Ref Expression
riotaiota  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )

Proof of Theorem riotaiota
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 iftrue 3745 . 2  |-  ( E! x  e.  A  ph  ->  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  ( iota x ( x  e.  A  /\  ph ) ) )
31, 2syl5eq 2480 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E!wreu 2707   ifcif 3739   iotacio 5416   ` cfv 5454   Undefcund 6541   iota_crio 6542
This theorem is referenced by:  riotauni  6556  riotacl2  6563  riota1  6568  riota2df  6570  snriota  6580  riotaprc  6587  ismgmid  14710  q1peqb  20077  adjval  23393
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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-if 3740  df-riota 6549
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