Theorem euor2 2272

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.xph -> (E!x(ph \/ ps) <-> E!xps))

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		nfe1 1732	. . 3 |- F/xE.xph
2	1	nfn 1793	. 2 |- F/x -. E.xph
3		19.8a 1756	. . . 4 |- (ph -> E.xph)
4	3	con3i 127	. . 3 |- (-. E.xph -> -. ph)
5		orel1 371	. . . 4 |- (-. ph -> ((ph \/ ps) -> ps))
6		olc 373	. . . 4 |- (ps -> (ph \/ ps))
7	5, 6	impbid1 194	. . 3 |- (-. ph -> ((ph \/ ps) <-> ps))
8	4, 7	syl 15	. 2 |- (-. E.xph -> ((ph \/ ps) <-> ps))
9	2, 8	eubid 2211	1 |- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357  E.wex 1541  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  reuun2  3539