MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eunex Structured version   Unicode version

Theorem eunex 4589
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 4587 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1655 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 180 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1745 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 466 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 eu3v 2270 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
7 exnal 1671 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
85, 6, 73imtr4i 268 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1405   E.wex 1635   E!weu 2240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-nul 4527  ax-pow 4574
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-eu 2244  df-mo 2245
This theorem is referenced by:  reusv2lem2  4598  unnt  30653  amosym1  30671  alneu  37587
  Copyright terms: Public domain W3C validator