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Theorem dfiota4 5512
Description: The  iota operation using the  if operator. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
dfiota4  |-  ( iota
x ph )  =  if ( E! x ph ,  U. { x  | 
ph } ,  (/) )

Proof of Theorem dfiota4
StepHypRef Expression
1 iotauni 5496 . . 3  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 iftrue 3900 . . 3  |-  ( E! x ph  ->  if ( E! x ph ,  U. { x  |  ph } ,  (/) )  = 
U. { x  | 
ph } )
31, 2eqtr4d 2496 . 2  |-  ( E! x ph  ->  ( iota x ph )  =  if ( E! x ph ,  U. { x  |  ph } ,  (/) ) )
4 iotanul 5499 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
5 iffalse 3902 . . 3  |-  ( -.  E! x ph  ->  if ( E! x ph ,  U. { x  | 
ph } ,  (/) )  =  (/) )
64, 5eqtr4d 2496 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  =  if ( E! x ph ,  U. { x  |  ph } ,  (/) ) )
73, 6pm2.61i 164 1  |-  ( iota
x ph )  =  if ( E! x ph ,  U. { x  | 
ph } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   E!weu 2261   {cab 2437   (/)c0 3740   ifcif 3894   U.cuni 4194   iotacio 5482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-uni 4195  df-iota 5484
This theorem is referenced by: (None)
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