Theorem iotaint 4900

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

Step	Hyp	Ref	Expression
1		iotauni 4899	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		uniintabim 3673	. 2 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
3	1, 2	eqtrd 2113	1 |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1284   E!weu 1941   {cab 2067   U.cuni 3601   |^|cint 3636   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-iota 4887

This theorem is referenced by:  bdcriota  10674