Theorem euor2 1999

Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
euor2	|- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		hbe1 1424	. . 3 |- ( E. x ph -> A. x E. x ph )
2	1	hbn 1584	. 2 |- ( -. E. x ph -> A. x -. E. x ph )
3		19.8a 1522	. . . 4 |- ( ph -> E. x ph )
4	3	con3i 594	. . 3 |- ( -. E. x ph -> -. ph )
5		orel1 676	. . . 4 |- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
6		olc 664	. . . 4 |- ( ps -> ( ph \/ ps ) )
7	5, 6	impbid1 140	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ps ) )
8	4, 7	syl 14	. 2 |- ( -. E. x ph -> ( ( ph \/ ps ) <-> ps ) )
9	2, 8	eubidh 1947	1 |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-eu 1944

This theorem is referenced by:  reuun2  3247