Theorem frege37 38134

Description: If ch is a necessary consequence of the occurrence of ps or ph, then ch is a necessary consequence of ph alone. Similar to a closed form of orcs 409. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege37	|- ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) )

Proof of Theorem frege37

Step	Hyp	Ref	Expression
1		frege36 38133	. 2 |- ( ph -> ( -. ph -> ps ) )
2		frege9 38106	. 2 |- ( ( ph -> ( -. ph -> ps ) ) -> ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ( -. ph -> ps ) -> ch ) -> ( ph -> ch ) )

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

This theorem is referenced by:  frege106  38263