Theorem 4exmid 905

Description: The disjunction of the four possible combinations of two wffs and their negations is always true. (Contributed by David Abernethy, 28-Jan-2014.)

Assertion

Ref	Expression
4exmid	|- (((ph /\ ps) \/ (-. ph /\ -. ps)) \/ ((ph /\ -. ps) \/ (ps /\ -. ph)))

Proof of Theorem 4exmid

Step	Hyp	Ref	Expression
1		exmid 404	. 2 |- ((ph <-> ps) \/ -. (ph <-> ps))
2		dfbi3 863	. . 3 |- ((ph <-> ps) <-> ((ph /\ ps) \/ (-. ph /\ -. ps)))
3		xor 861	. . 3 |- (-. (ph <-> ps) <-> ((ph /\ -. ps) \/ (ps /\ -. ph)))
4	2, 3	orbi12i 507	. 2 |- (((ph <-> ps) \/ -. (ph <-> ps)) <-> (((ph /\ ps) \/ (-. ph /\ -. ps)) \/ ((ph /\ -. ps) \/ (ps /\ -. ph))))
5	1, 4	mpbi 199	1 |- (((ph /\ ps) \/ (-. ph /\ -. ps)) \/ ((ph /\ -. ps) \/ (ps /\ -. ph)))

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

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

This theorem is referenced by: (None)