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.

Step	Hyp	Ref	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 ) )
3	1, 2	mtoi 182	. . . 4 |- ( A. x ( ph -> x = y ) -> -. A. x ph )
4	3	exlimiv 1767	. . 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 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 )
8	5, 6, 7	3imtr4i 270	1 |- ( E! x ph -> E. x -. ph )

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