Theorem eunex 4859

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 4857	. . . . 5 |- -. A. x x = y
2		alim 1738	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( A. x ph -> A. x x = y ) )
3	1, 2	mtoi 190	. . . 4 |- ( A. x ( ph -> x = y ) -> -. A. x ph )
4	3	exlimiv 1858	. . 3 |- ( E. y A. x ( ph -> x = y ) -> -. A. x ph )
5	4	adantl 482	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> -. A. x ph )
6		eu3v 2498	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
7		exnal 1754	. 2 |- ( E. x -. ph <-> -. A. x ph )
8	5, 6, 7	3imtr4i 281	1 |- ( E! x ph -> E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  reusv2lem2  4869  reusv2lem2OLD  4870  unnt  32407  amosym1  32425  alneu  41201