Theorem iotauni 5299

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 2082	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		iotaval 5298	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
3		uniabio 5297	. . . 4 |- ( A. x ( ph <-> x = z ) -> U. { x | ph } = z )
4	2, 3	eqtr4d 2267	. . 3 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
5	4	exlimiv 1646	. 2 |- ( E. z A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
6	1, 5	sylbi 121	1 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   = wceq 1397   E.wex 1540   E!weu 2079   {cab 2217   U.cuni 3893   iotacio 5284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286

This theorem is referenced by:  iotaint  5300  fveu  5631  riotauni  5977