Theorem euor2 2514

Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Assertion

Ref	Expression
euor2	|- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Proof of Theorem euor2

Step	Hyp	Ref	Expression
1		nfe1 2027	. . 3 |- F/ x E. x ph
2	1	nfn 1784	. 2 |- F/ x -. E. x ph
3		19.8a 2052	. . . 4 |- ( ph -> E. x ph )
4	3	con3i 150	. . 3 |- ( -. E. x ph -> -. ph )
5		biorf 420	. . . 4 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
6	5	bicomd 213	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ps ) )
7	4, 6	syl 17	. 2 |- ( -. E. x ph -> ( ( ph \/ ps ) <-> ps ) )
8	2, 7	eubid 2488	1 |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by:  reuun2  3910