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Theorem 2euswapdc 2129
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 1674 . . . . 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 2128 . . . . 5  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) ) )
43imp 124 . . . 4  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
52, 4anim12d 335 . . 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 2085 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
7 eu5 2085 . . 3  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
85, 6, 73imtr4g 205 . 2  |-  ( (DECID  E. x E. y ph  /\ 
A. x E* y ph )  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
98ex 115 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 104  DECID wdc 835   A.wal 1362   E.wex 1503   E!weu 2038   E*wmo 2039
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042
This theorem is referenced by:  euxfr2dc  2937  2reuswapdc  2956
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