Theorem frege48

Description: Closed form of syllogism with internal disjunction. If ph is a sufficient condition for the occurrence of ch or ps and if ch , as well as ps , is a sufficient condition for th , then ph is a sufficient condition for th . See application in frege101 . Proposition 48 of Frege1879 p. 49. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege48	|- ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege47	|- ( ( -. ps -> ch ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> th ) ) )
2		frege23	|- ( ( ( -. ps -> ch ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> th ) ) ) -> ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) ) )
3	1 2	ax-mp	|- ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) )