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

Theorem 2euswapdc 2066
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 1624 . . . . 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 2065 . . . . 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 331 . . 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 2022 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
7 eu5 2022 . . 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 802   A.wal 1312   E.wex 1451   E!weu 1975   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:  euxfr2dc  2840  2reuswapdc  2859
  Copyright terms: Public domain W3C validator