Theorem 3anor 948

Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)

Assertion

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

Proof of Theorem 3anor

Step	Hyp	Ref	Expression
1		df-3an 936	. 2 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2		anor 475	. . . 4 |- (((ph /\ ps) /\ ch) <-> -. (-. (ph /\ ps) \/ -. ch))
3		ianor 474	. . . . 5 |- (-. (ph /\ ps) <-> (-. ph \/ -. ps))
4	3	orbi1i 506	. . . 4 |- ((-. (ph /\ ps) \/ -. ch) <-> ((-. ph \/ -. ps) \/ -. ch))
5	2, 4	xchbinx 301	. . 3 |- (((ph /\ ps) /\ ch) <-> -. ((-. ph \/ -. ps) \/ -. ch))
6		df-3or 935	. . 3 |- ((-. ph \/ -. ps \/ -. ch) <-> ((-. ph \/ -. ps) \/ -. ch))
7	5, 6	xchbinxr 302	. 2 |- (((ph /\ ps) /\ ch) <-> -. (-. ph \/ -. ps \/ -. ch))
8	1, 7	bitri 240	1 |- ((ph /\ ps /\ ch) <-> -. (-. ph \/ -. ps \/ -. ch))

Syntax hints:  -. wn 3   <-> wb 176   \/ wo 357   /\ wa 358   \/ w3o 933   /\ w3a 934

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 177  df-or 359  df-an 360  df-3or 935  df-3an 936

This theorem is referenced by:  3ianor  949  ne3anior  2603