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Theorem eunex 4377
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
eunex  |-  ( E! x ph  ->  E. x  -.  ph )

Proof of Theorem eunex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . . 3  |-  F/ y
ph
21eu3 1994 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
3 dtruex 4375 . . . . 5  |-  E. x  -.  x  =  y
4 nfa1 1479 . . . . . 6  |-  F/ x A. x ( ph  ->  x  =  y )
5 sp 1446 . . . . . . 7  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  x  =  y ) )
65con3d 596 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( -.  x  =  y  ->  -.  ph )
)
74, 6eximd 1548 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x  -.  x  =  y  ->  E. x  -.  ph )
)
83, 7mpi 15 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  E. x  -.  ph )
98exlimiv 1534 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  E. x  -.  ph )
109adantl 271 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  E. x  -.  ph )
112, 10sylbi 119 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289   E.wex 1426   E!weu 1948
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452
This theorem is referenced by: (None)
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