Theorem euiotaex 4826

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!xph -> ( iotaxph) e. _V)

Proof of Theorem euiotaex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 4821	. . . 4 |- (A.x(ph <-> x = y) -> ( iotaxph) = y)
2	1	eqcomd 2042	. . 3 |- (A.x(ph <-> x = y) -> y = ( iotaxph))
3	2	eximi 1488	. 2 |- (E.yA.x(ph <-> x = y) -> E.y y = ( iotaxph))
4		df-eu 1900	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
5		isset 2555	. 2 |- (( iotaxph) e. _V <-> E.y y = ( iotaxph))
6	3, 4, 5	3imtr4i 190	1 |- (E!xph -> ( iotaxph) e. _V)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390  E!weu 1897   _Vcvv 2551   iotacio 4808

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810

This theorem is referenced by:  iota4an  4829  funfvex  5135