Theorem frege9 38106

Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 38094 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege9	|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )

Proof of Theorem frege9

Step	Hyp	Ref	Expression
1		frege5 38094	. 2 |- ( ( ps -> ch ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
2		ax-frege8 38103	. 2 |- ( ( ( ps -> ch ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> 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:  frege11  38108  frege10  38114  frege19  38118  frege21  38121  frege37  38134  frege56aid  38164  frege56a  38165  frege61a  38173  frege56b  38192  frege61b  38200  frege56c  38213  frege61c  38218  frege117  38274  frege130  38287  frege132  38289