ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2moswapdc Unicode version

Theorem 2moswapdc 2035
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 1428 . . . 4  |-  F/ y E. y ph
21moexexdc 2029 . . 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 1373 . 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 1525 . . . . . 6  |-  ( ph  ->  E. y ph )
54pm4.71ri 384 . . . . 5  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1539 . . . 4  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 1982 . . 3  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
87imbi2i 224 . 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 160 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 778   A.wal 1285   E.wex 1424   E*wmo 1946
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-dc 779  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by:  2euswapdc  2036
  Copyright terms: Public domain W3C validator