Theorem euor2 2006

Description: Introduce or eliminate a disjunct in a unique existential 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 1429	. . 3 |- ( E. x ph -> A. x E. x ph )
2	1	hbn 1589	. 2 |- ( -. E. x ph -> A. x -. E. x ph )
3		19.8a 1527	. . . 4 |- ( ph -> E. x ph )
4	3	con3i 597	. . 3 |- ( -. E. x ph -> -. ph )
5		orel1 679	. . . 4 |- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
6		olc 667	. . . 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 1954	1 |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 664   E.wex 1426   E!weu 1948

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-eu 1951

This theorem is referenced by:  reuun2  3282