Metamath Proof Explorer

Theorem frege59a

Description: A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of Frege1879 p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege59a	\|- ( if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege58acor	\|- ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )
2		frege30	\|- ( ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) ) -> ( if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) ) )
3	1 2	ax-mp	\|- ( if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) )

Step

Hyp

Ref

Expression

frege58acor

 |-  ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )

frege30

 |-  ( ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) ) -> ( if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) ) )

1 2

ax-mp

 |-  ( if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) )