Theorem eliotaeu 5224

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 1629	. 2 |- ( E. y ( A e. y /\ A. x ( ph <-> x = y ) ) -> E. y A. x ( ph <-> x = y ) )
2		eliota 5223	. 2 |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
3		df-eu 2041	. 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 1362   E.wex 1503   E!weu 2038   e. wcel 2160   iotacio 5194

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sn 3613  df-uni 3825  df-iota 5196

This theorem is referenced by:  iotam  5227  elfvm  5567