Theorem frege46 38144

Description: If ps holds when ph occurs as well as when ph does not occur, then ps holds. If ps or ph occurs and if the occurences of ph has ps as a necessary consequence, then ps takes place. Identical to pm2.6 182. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege46	|- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) )

Proof of Theorem frege46

Step	Hyp	Ref	Expression
1		frege33 38130	. 2 |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) )
2		frege45 38143	. 2 |- ( ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )
3	1, 2	ax-mp 5	1 |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) )

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:  frege47  38145