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Theorem iotaint 4880
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 4879 . 2  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 uniintabim 3652 . 2  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
31, 2eqtrd 2072 1  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   E!weu 1900   {cab 2026   U.cuni 3580   |^|cint 3615   iotacio 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-iota 4867
This theorem is referenced by:  bdcriota  10003
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