Theorem 3ianor

Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Andrew Salmon, 13-May-2011) Shorten with xchnxbir . (Revised by Wolf Lammen, 8-Apr-2022)

	Ref	Expression
Assertion	3ianor	|- ( -. ( ph /\ ps /\ ch ) <-> ( -. ph \/ -. ps \/ -. ch ) )

Proof

Step	Hyp	Ref	Expression
1		ianor	|- ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) )
2	1	orbi1i	|- ( ( -. ( ph /\ ps ) \/ -. ch ) <-> ( ( -. ph \/ -. ps ) \/ -. ch ) )
3		ianor	|- ( -. ( ( ph /\ ps ) /\ ch ) <-> ( -. ( ph /\ ps ) \/ -. ch ) )
4		df-3an	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
5	3 4	xchnxbir	|- ( -. ( ph /\ ps /\ ch ) <-> ( -. ( ph /\ ps ) \/ -. ch ) )
6		df-3or	|- ( ( -. ph \/ -. ps \/ -. ch ) <-> ( ( -. ph \/ -. ps ) \/ -. ch ) )
7	2 5 6	3bitr4i	|- ( -. ( ph /\ ps /\ ch ) <-> ( -. ph \/ -. ps \/ -. ch ) )