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Theorem 2euswapdc 2068
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
Assertion
Ref Expression
2euswapdc  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) ) )

Proof of Theorem 2euswapdc
StepHypRef Expression
1 excomim 1626 . . . . 5  |-  ( E. x E. y ph  ->  E. y E. x ph )
21a1i 9 . . . 4  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E. x E. y ph  ->  E. y E. x ph ) )
3 2moswapdc 2067 . . . . 5  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) ) )
43imp 123 . . . 4  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
52, 4anim12d 333 . . 3  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  ( E. y E. x ph  /\ 
E* y E. x ph ) ) )
6 eu5 2024 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
7 eu5 2024 . . 3  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
85, 6, 73imtr4g 204 . 2  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
98ex 114 1  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 804   A.wal 1314   E.wex 1453   E!weu 1977   E*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by:  euxfr2dc  2842  2reuswapdc  2861
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