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

Theorem eunex 4352
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 4350 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1564 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 171 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1641 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 453 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 nfv 1626 . . 3  |-  F/ y
ph
76eu3 2280 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
8 exnal 1580 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
95, 7, 83imtr4i 258 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547   E!weu 2254
This theorem is referenced by:  reusv2lem2  4684  unnt  26062  amosym1  26080  alneu  27846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-nul 4298  ax-pow 4337
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258
  Copyright terms: Public domain W3C validator