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.

Step	Hyp	Ref	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 ) )
3	1, 2	mtoi 180	. . . 4 |- ( A. x ( ph -> x = y ) -> -. A. x ph )
4	3	exlimiv 1745	. . 3 |- ( E. y A. x ( ph -> x = y ) -> -. A. x ph )
5	4	adantl 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 )
8	5, 6, 7	3imtr4i 268	1 |- ( E! x ph -> E. x -. ph )

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