Theorem euiotaex 4883

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 4878	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2045	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1491	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 1903	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 2561	. 2 |- ( ( iota x ph ) e. _V <-> E. y y = ( iota x ph ) )
6	3, 4, 5	3imtr4i 190	1 |- ( E! x ph -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   _Vcvv 2557   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867

This theorem is referenced by:  iota4an  4886  funfvex  5192