Theorem euor2 1958

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 1384	. . 3 |- ( E. x ph -> A. x E. x ph )
2	1	hbn 1544	. 2 |- ( -. E. x ph -> A. x -. E. x ph )
3		19.8a 1482	. . . 4 |- ( ph -> E. x ph )
4	3	con3i 562	. . 3 |- ( -. E. x ph -> -. ph )
5		orel1 644	. . . 4 |- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
6		olc 632	. . . 4 |- ( ps -> ( ph \/ ps ) )
7	5, 6	impbid1 130	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ps ) )
8	4, 7	syl 14	. 2 |- ( -. E. x ph -> ( ( ph \/ ps ) <-> ps ) )
9	2, 8	eubidh 1906	1 |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-eu 1903

This theorem is referenced by:  reuun2  3220