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Theorem 2euswapdc 2033
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 1594 . . . . 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 2032 . . . . 5  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) ) )
43imp 122 . . . 4  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
52, 4anim12d 328 . . 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 1989 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
7 eu5 1989 . . 3  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
85, 6, 73imtr4g 203 . 2  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
98ex 113 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 102  DECID wdc 776   A.wal 1283   E.wex 1422   E!weu 1942   E*wmo 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by:  euxfr2dc  2778  2reuswapdc  2795
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