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Theorem 2moswapdc 2065
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
Assertion
Ref Expression
2moswapdc  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) ) )

Proof of Theorem 2moswapdc
StepHypRef Expression
1 nfe1 1455 . . . 4  |-  F/ y E. y ph
21moexexdc 2059 . . 3  |-  (DECID  E. x E. y ph  ->  (
( E* x E. y ph  /\  A. x E* y ph )  ->  E* y E. x ( E. y ph  /\  ph ) ) )
32expcomd 1400 . 2  |-  (DECID  E. x E. y ph  ->  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) ) )
4 19.8a 1552 . . . . . 6  |-  ( ph  ->  E. y ph )
54pm4.71ri 387 . . . . 5  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1567 . . . 4  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2012 . . 3  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
87imbi2i 225 . 2  |-  ( ( E* x E. y ph  ->  E* y E. x ph )  <->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) )
93, 8syl6ibr 161 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 802   A.wal 1312   E.wex 1451   E*wmo 1976
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by:  2euswapdc  2066
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