Theorem iotauni 5146

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 2009	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		iotaval 5145	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
3		uniabio 5144	. . . 4 |- ( A. x ( ph <-> x = z ) -> U. { x | ph } = z )
4	2, 3	eqtr4d 2193	. . 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 1333   = wceq 1335   E.wex 1472   E!weu 2006   {cab 2143   U.cuni 3772   iotacio 5132

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3566  df-pr 3567  df-uni 3773  df-iota 5134

This theorem is referenced by:  iotaint  5147  fveu  5459  riotauni  5783