Theorem df3or2

Description: Express triple-or in terms of implication and negation. Statement in Frege1879 p. 11. (Contributed by RP, 25-Jul-2020)

	Ref	Expression
Assertion	df3or2	|- ( ( ph \/ ps \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-3or	|- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
2		df-or	|- ( ( ( ph \/ ps ) \/ ch ) <-> ( -. ( ph \/ ps ) -> ch ) )
3		ioran	|- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
4	3	imbi1i	|- ( ( -. ( ph \/ ps ) -> ch ) <-> ( ( -. ph /\ -. ps ) -> ch ) )
5		impexp	|- ( ( ( -. ph /\ -. ps ) -> ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
6	4 5	bitri	|- ( ( -. ( ph \/ ps ) -> ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
7	2 6	bitri	|- ( ( ( ph \/ ps ) \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
8	1 7	bitri	|- ( ( ph \/ ps \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )