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Theorem eliota 5203
Description: An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
Assertion
Ref Expression
eliota  |-  ( A  e.  ( iota x ph )  <->  E. y ( A  e.  y  /\  A. x ( ph  <->  x  =  y ) ) )
Distinct variable groups:    ph, y    y, A    x, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem eliota
StepHypRef Expression
1 dfiota2 5178 . . 3  |-  ( iota
x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
21eleq2i 2244 . 2  |-  ( A  e.  ( iota x ph )  <->  A  e.  U. {
y  |  A. x
( ph  <->  x  =  y
) } )
3 eluniab 3821 . 2  |-  ( A  e.  U. { y  |  A. x (
ph 
<->  x  =  y ) }  <->  E. y ( A  e.  y  /\  A. x ( ph  <->  x  =  y ) ) )
42, 3bitri 184 1  |-  ( A  e.  ( iota x ph )  <->  E. y ( A  e.  y  /\  A. x ( ph  <->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   A.wal 1351   E.wex 1492    e. wcel 2148   {cab 2163   U.cuni 3809   iotacio 5175
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sn 3598  df-uni 3810  df-iota 5177
This theorem is referenced by:  eliotaeu  5204
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