Theorem eliotaeu 5247

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 1632	. 2 |- ( E. y ( A e. y /\ A. x ( ph <-> x = y ) ) -> E. y A. x ( ph <-> x = y ) )
2		eliota 5246	. 2 |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
3		df-eu 2048	. 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 1506   E!weu 2045   e. wcel 2167   iotacio 5217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sn 3628  df-uni 3840  df-iota 5219

This theorem is referenced by:  iotam  5250  elfvm  5591