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Theorem eunex 4332
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 1462 . . 3  |-  F/ y
ph
21eu3 1989 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
3 dtruex 4330 . . . . 5  |-  E. x  -.  x  =  y
4 nfa1 1475 . . . . . 6  |-  F/ x A. x ( ph  ->  x  =  y )
5 sp 1442 . . . . . . 7  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  x  =  y ) )
65con3d 594 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( -.  x  =  y  ->  -.  ph )
)
74, 6eximd 1544 . . . . 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 1530 . . 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 1283    = wceq 1285   E.wex 1422   E!weu 1943
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422
This theorem is referenced by: (None)
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