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Theorem iotaint 5503
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 5502 . 2  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
2 uniintab 4275 . . 3  |-  ( E! x ph  <->  U. { x  |  ph }  =  |^| { x  |  ph }
)
32biimpi 194 . 2  |-  ( E! x ph  ->  U. {
x  |  ph }  =  |^| { x  | 
ph } )
41, 3eqtrd 2495 1  |-  ( E! x ph  ->  ( iota x ph )  = 
|^| { x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   E!weu 2262   {cab 2439   U.cuni 4200   |^|cint 4237   iotacio 5488
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 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-sn 3987  df-pr 3989  df-uni 4201  df-int 4238  df-iota 5490
This theorem is referenced by: (None)
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