Theorem eliotaeu 5279

Description: An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)

Assertion

Ref	Expression
eliotaeu	|- ( A e. ( iota x ph ) -> E! x ph )

Proof of Theorem eliotaeu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpr 1642	. 2 |- ( E. y ( A e. y /\ A. x ( ph <-> x = y ) ) -> E. y A. x ( ph <-> x = y ) )
2		eliota 5278	. 2 |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
3		df-eu 2058	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
4	1, 2, 3	3imtr4i 201	1 |- ( A e. ( iota x ph ) -> E! x ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   E.wex 1516   E!weu 2055   e. wcel 2178   iotacio 5249

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sn 3649  df-uni 3865  df-iota 5251

This theorem is referenced by:  iotam  5282  elfvm  5632