Theorem iotaexeu 38619

Description: The iota class exists. This theorem does not require ax-nul 4789 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaexeu	|- ( E! x ph -> ( iota x ph ) e. _V )

Proof of Theorem iotaexeu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 5862	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2	1	eqcomd 2628	. . 3 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
3	2	eximi 1762	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y y = ( iota x ph ) )
4		df-eu 2474	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
5		isset 3207	. 2 |- ( ( iota x ph ) e. _V <-> E. y y = ( iota x ph ) )
6	3, 4, 5	3imtr4i 281	1 |- ( E! x ph -> ( iota x ph ) e. _V )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  iotasbc  38620  pm14.18  38629  iotavalb  38631  sbiota1  38635