Theorem pm5.17 932

Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Assertion

Ref	Expression
pm5.17	|- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) )

Proof of Theorem pm5.17

Step	Hyp	Ref	Expression
1		bicom 212	. 2 |- ( ( ph <-> -. ps ) <-> ( -. ps <-> ph ) )
2		dfbi2 660	. 2 |- ( ( -. ps <-> ph ) <-> ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) )
3		orcom 402	. . . 4 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
4		df-or 385	. . . 4 |- ( ( ps \/ ph ) <-> ( -. ps -> ph ) )
5	3, 4	bitr2i 265	. . 3 |- ( ( -. ps -> ph ) <-> ( ph \/ ps ) )
6		imnan 438	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
7	5, 6	anbi12i 733	. 2 |- ( ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
8	1, 2, 7	3bitrri 287	1 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by:  nbi2  936  odd2np1  15065  ordtconnlem1  29970  sgnneg  30602