Theorem iotauni 5243

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 2056	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		iotaval 5242	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
3		uniabio 5241	. . . 4 |- ( A. x ( ph <-> x = z ) -> U. { x | ph } = z )
4	2, 3	eqtr4d 2240	. . 3 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
5	4	exlimiv 1620	. 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 1370   = wceq 1372   E.wex 1514   E!weu 2053   {cab 2190   U.cuni 3849   iotacio 5229

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850  df-iota 5231

This theorem is referenced by:  iotaint  5244  fveu  5567  riotauni  5905