Theorem frege18 38112

Description: Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege18	|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( ps -> ( th -> ch ) ) ) )

Proof of Theorem frege18

Step	Hyp	Ref	Expression
1		frege5 38094	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( th -> ( ps -> ch ) ) ) )
2		frege16 38110	. 2 |- ( ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( th -> ( ps -> ch ) ) ) ) -> ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( ps -> ( th -> ch ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ph ) -> ( ps -> ( th -> ch ) ) ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103

This theorem is referenced by:  frege19  38118  frege23  38119  frege20  38122  frege51  38149  frege64a  38176  frege64b  38203  frege64c  38221  frege82  38239