Theorem frege47 38145

Description: Deduce consequence follows from either path implied by a disjunction. If ph, as well as ps is sufficient condition for ch and ps or ph takes place, then the proposition ch holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege47	|- ( ( -. ph -> ps ) -> ( ( ps -> ch ) -> ( ( ph -> ch ) -> ch ) ) )

Proof of Theorem frege47

Step	Hyp	Ref	Expression
1		frege46 38144	. 2 |- ( ( -. ph -> ch ) -> ( ( ph -> ch ) -> ch ) )
2		frege21 38121	. 2 |- ( ( ( -. ph -> ch ) -> ( ( ph -> ch ) -> ch ) ) -> ( ( -. ph -> ps ) -> ( ( ps -> ch ) -> ( ( ph -> ch ) -> ch ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( -. ph -> ps ) -> ( ( ps -> ch ) -> ( ( ph -> ch ) -> 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  ax-frege41 38139

This theorem is referenced by:  frege48  38146  frege49  38147