Theorem euor2 2103

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 1509	. . 3 |- ( E. x ph -> A. x E. x ph )
2	1	hbn 1668	. 2 |- ( -. E. x ph -> A. x -. E. x ph )
3		19.8a 1604	. . . 4 |- ( ph -> E. x ph )
4	3	con3i 633	. . 3 |- ( -. E. x ph -> -. ph )
5		orel1 726	. . . 4 |- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
6		olc 712	. . . 4 |- ( ps -> ( ph \/ ps ) )
7	5, 6	impbid1 142	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ps ) )
8	4, 7	syl 14	. 2 |- ( -. E. x ph -> ( ( ph \/ ps ) <-> ps ) )
9	2, 8	eubidh 2051	1 |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   E.wex 1506   E!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-eu 2048

This theorem is referenced by:  reuun2  3446