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Theorem 2euex 2123
Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex  |-  ( E! x E. y ph  ->  E. y E! x ph )

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2083 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1674 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 hbe1 1505 . . . . . 6  |-  ( E. y ph  ->  A. y E. y ph )
43hbmo 2075 . . . . 5  |-  ( E* x E. y ph  ->  A. y E* x E. y ph )
5 19.8a 1600 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2101 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 2040 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 122 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximdh 1621 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9biimtrid 152 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 125 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 121 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1502   E!weu 2036   E*wmo 2037
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040
This theorem is referenced by: (None)
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