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Theorem 2moswapdc 2116
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 1496 . . . 4  |-  F/ y E. y ph
21moexexdc 2110 . . 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 1441 . 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 1590 . . . . . 6  |-  ( ph  ->  E. y ph )
54pm4.71ri 392 . . . . 5  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1605 . . . 4  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2063 . . 3  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
87imbi2i 226 . 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 162 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 834   A.wal 1351   E.wex 1492   E*wmo 2027
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by:  2euswapdc  2117
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