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Theorem iotaint 5434
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 5433 . 2  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 uniintab 4090 . . 3  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
32biimpi 188 . 2  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
41, 3eqtrd 2470 1  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   E!weu 2283   {cab 2424   U.cuni 4017   |^|cint 4052   iotacio 5419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-uni 4018  df-int 4053  df-iota 5421
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