Theorem eunex 4595

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 4593	. . . . 5 |- -. A. x x = y
2		alim 1682	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( A. x ph -> A. x x = y ) )
3	1, 2	mtoi 182	. . . 4 |- ( A. x ( ph -> x = y ) -> -. A. x ph )
4	3	exlimiv 1775	. . 3 |- ( E. y A. x ( ph -> x = y ) -> -. A. x ph )
5	4	adantl 468	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> -. A. x ph )
6		eu3v 2326	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
7		exnal 1698	. 2 |- ( E. x -. ph <-> -. A. x ph )
8	5, 6, 7	3imtr4i 270	1 |- ( E! x ph -> E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   A.wal 1441   E.wex 1662   E!weu 2298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-nul 4533  ax-pow 4580

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-eu 2302  df-mo 2303

This theorem is referenced by:  reusv2lem2  4604  unnt  31061  amosym1  31079  alneu  38616