Theorem iotauni 5108

Description: Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iotauni	|- ( E! x ph -> ( iota x ph ) = U. { x | ph } )

Proof of Theorem iotauni

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2003	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		iotaval 5107	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
3		uniabio 5106	. . . 4 |- ( A. x ( ph <-> x = z ) -> U. { x | ph } = z )
4	2, 3	eqtr4d 2176	. . 3 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
5	4	exlimiv 1578	. 2 |- ( E. z A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
6	1, 5	sylbi 120	1 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   E!weu 2000   {cab 2126   U.cuni 3744   iotacio 5094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096

This theorem is referenced by:  iotaint  5109  fveu  5421  riotauni  5744