Theorem euiotaex 5352

Description: Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)

Assertion

Ref	Expression
euiotaex	|- ( E! x ph -> ( iota x ph ) e. _V )

Proof of Theorem euiotaex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 5347	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2244	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1653	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 2089	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 2828	. 2 |- ( ( iota x ph ) e. _V <-> E. y y = ( iota x ph ) )
6	3, 4, 5	3imtr4i 201	1 |- ( E! x ph -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   _Vcvv 2821   iotacio 5333

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3714  df-pr 3715  df-uni 3934  df-iota 5335

This theorem is referenced by:  iota4an  5356  funfvex  5710  elfvfvex  5727  pcval  13058  gsumvalfi  14135