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Theorem riotaiota 6326
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 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 iftrue 3584 . 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 2340 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 358    = wceq 1632    e. wcel 1696   {cab 2282   E!wreu 2558   ifcif 3578   iotacio 5233   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem is referenced by:  riotauni  6327  riotacl2  6334  riota1  6339  riota2df  6341  snriota  6351  riotaprc  6358  ismgmid  14403  q1peqb  19556  adjval  22486
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579  df-riota 6320
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