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Theorem 2euswapdc 2097
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 1643 . . . . 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 2096 . . . . 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 2053 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
7 eu5 2053 . . 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 820   A.wal 1333   E.wex 1472   E!weu 2006   E*wmo 2007
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010
This theorem is referenced by:  euxfr2dc  2897  2reuswapdc  2916
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