Theorem euiotaex 5328

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 5323	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2238	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1649	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 2083	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 2819	. 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 1396   = wceq 1398   E.wex 1541   E!weu 2080   e. wcel 2203   _Vcvv 2812   iotacio 5309

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-sn 3694  df-pr 3695  df-uni 3914  df-iota 5311

This theorem is referenced by:  iota4an  5332  funfvex  5686  elfvfvex  5703  pcval  12987  gfsumval  16848