Theorem frege48 38146

Description: Closed form of syllogism with internal disjunction. If ph is a sufficient condition for the occurence 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 38258. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege48

Step	Hyp	Ref	Expression
1		frege47 38145	. 2 |- ( ( -. ps -> ch ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> th ) ) )
2		frege23 38119	. 2 |- ( ( ( -. ps -> ch ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> th ) ) ) -> ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ph -> ( -. ps -> ch ) ) -> ( ( ch -> th ) -> ( ( ps -> th ) -> ( ph -> th ) ) ) )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege28 38124  ax-frege31 38128  ax-frege41 38139

This theorem is referenced by:  frege101  38258