Theorem euor2 1839

Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (The proof was 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		hbe1 1363	. . 3 |- (E.xph -> A.xE.xph)
2	1	hbn 1351	. 2 |- (-. E.xph -> A.x -. E.xph)
3		19.8a 1376	. . . 4 |- (ph -> E.xph)
4	3	con3i 114	. . 3 |- (-. E.xph -> -. ph)
5		orel1 271	. . . 4 |- (-. ph -> ((ph \/ ps) -> ps))
6		olc 290	. . . 4 |- (ps -> (ph \/ ps))
7	5, 6	impbid1 575	. . 3 |- (-. ph -> ((ph \/ ps) <-> ps))
8	4, 7	syl 12	. 2 |- (-. E.xph -> ((ph \/ ps) <-> ps))
9	2, 8	eubid 1778	1 |- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239  E.wex 1326  E!weu 1771

This theorem is referenced by:  reuun2 2873  bnj111OLD 12456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-eu 1775

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