Theorem euiotaex 5186

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 5181	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2181	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1598	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 2027	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 2741	. 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 1351   = wceq 1353   E.wex 1490   E!weu 2024   e. wcel 2146   _Vcvv 2735   iotacio 5168

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-sn 3595  df-pr 3596  df-uni 3806  df-iota 5170

This theorem is referenced by:  iota4an  5189  funfvex  5524  pcval  12263