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Theorem iotaint 5335
Description: Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
iotaint  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )

Proof of Theorem iotaint
StepHypRef Expression
1 iotauni 5334 . 2  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 uniintab 4002 . . 3  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
32biimpi 186 . 2  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
41, 3eqtrd 2398 1  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647   E!weu 2217   {cab 2352   U.cuni 3929   |^|cint 3964   iotacio 5320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-sn 3735  df-pr 3736  df-uni 3930  df-int 3965  df-iota 5322
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