ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euxfr2dc Unicode version

Theorem euxfr2dc 2842
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
euxfr2dc.1  |-  A  e. 
_V
euxfr2dc.2  |-  E* y  x  =  A
Assertion
Ref Expression
euxfr2dc  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph ) )
Distinct variable groups:    ph, x    x, A
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem euxfr2dc
StepHypRef Expression
1 euxfr2dc.2 . . . . . . 7  |-  E* y  x  =  A
21moani 2047 . . . . . 6  |-  E* y
( ph  /\  x  =  A )
3 ancom 264 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  <->  ( x  =  A  /\  ph )
)
43mobii 2014 . . . . . 6  |-  ( E* y ( ph  /\  x  =  A )  <->  E* y ( x  =  A  /\  ph )
)
52, 4mpbi 144 . . . . 5  |-  E* y
( x  =  A  /\  ph )
65ax-gen 1410 . . . 4  |-  A. x E* y ( x  =  A  /\  ph )
7 excom 1627 . . . . . 6  |-  ( E. y E. x ( x  =  A  /\  ph )  <->  E. x E. y
( x  =  A  /\  ph ) )
87dcbii 810 . . . . 5  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  <-> DECID  E. x E. y ( x  =  A  /\  ph )
)
9 2euswapdc 2068 . . . . 5  |-  (DECID  E. x E. y ( x  =  A  /\  ph )  ->  ( A. x E* y ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) ) )
108, 9sylbi 120 . . . 4  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( A. x E* y ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) ) )
116, 10mpi 15 . . 3  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  ->  E! y E. x
( x  =  A  /\  ph ) ) )
12 moeq 2832 . . . . . . 7  |-  E* x  x  =  A
1312moani 2047 . . . . . 6  |-  E* x
( ph  /\  x  =  A )
143mobii 2014 . . . . . 6  |-  ( E* x ( ph  /\  x  =  A )  <->  E* x ( x  =  A  /\  ph )
)
1513, 14mpbi 144 . . . . 5  |-  E* x
( x  =  A  /\  ph )
1615ax-gen 1410 . . . 4  |-  A. y E* x ( x  =  A  /\  ph )
17 2euswapdc 2068 . . . 4  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( A. y E* x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y
( x  =  A  /\  ph ) ) ) )
1816, 17mpi 15 . . 3  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! y E. x ( x  =  A  /\  ph )  ->  E! x E. y
( x  =  A  /\  ph ) ) )
1911, 18impbid 128 . 2  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y E. x ( x  =  A  /\  ph ) ) )
20 euxfr2dc.1 . . . 4  |-  A  e. 
_V
21 biidd 171 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
ph ) )
2220, 21ceqsexv 2699 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ph )
2322eubii 1986 . 2  |-  ( E! y E. x ( x  =  A  /\  ph )  <->  E! y ph )
2419, 23syl6bb 195 1  |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 804   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   E!weu 1977   E*wmo 1978   _Vcvv 2660
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  euxfrdc  2843
  Copyright terms: Public domain W3C validator