Theorem pm5.17

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

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

Proof

Step	Hyp	Ref	Expression
1		bicom	|- ( ( ph <-> -. ps ) <-> ( -. ps <-> ph ) )
2		dfbi2	|- ( ( -. ps <-> ph ) <-> ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) )
3		orcom	|- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
4		df-or	|- ( ( ps \/ ph ) <-> ( -. ps -> ph ) )
5	3 4	bitr2i	|- ( ( -. ps -> ph ) <-> ( ph \/ ps ) )
6		imnan	|- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
7	5 6	anbi12i	|- ( ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
8	1 2 7	3bitrri	|- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) )