Theorem iotauni 5192

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 2029	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		iotaval 5191	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
3		uniabio 5190	. . . 4 |- ( A. x ( ph <-> x = z ) -> U. { x | ph } = z )
4	2, 3	eqtr4d 2213	. . 3 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = U. { x | ph } )
5	4	exlimiv 1598	. 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 1351   = wceq 1353   E.wex 1492   E!weu 2026   {cab 2163   U.cuni 3811   iotacio 5178

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812  df-iota 5180

This theorem is referenced by:  iotaint  5193  fveu  5509  riotauni  5839