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Theorem eunex 4615
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)
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 dtru 4613 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1680 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 182 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1767 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 468 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 eu3v 2295 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
7 exnal 1696 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
85, 6, 73imtr4i 270 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371   A.wal 1436   E.wex 1660   E!weu 2266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-nul 4553  ax-pow 4600
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-eu 2270  df-mo 2271
This theorem is referenced by:  reusv2lem2  4624  unnt  31067  amosym1  31085  alneu  38341
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