Theorem euiotaex 5040

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 5035	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2105	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1547	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 1963	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 2647	. 2 |- ( ( iota x ph ) e. _V <-> E. y y = ( iota x ph ) )
6	3, 4, 5	3imtr4i 200	1 |- ( E! x ph -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1297   = wceq 1299   E.wex 1436   e. wcel 1448   E!weu 1960   _Vcvv 2641   iotacio 5022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-sn 3480  df-pr 3481  df-uni 3684  df-iota 5024

This theorem is referenced by:  iota4an  5043  funfvex  5370