Metamath Proof Explorer

Theorem frege62a

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a particular context. Proposition 62 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege62a	\|- ( if- ( ph , ps , th ) -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege58acor	\|- ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )
2		ax-frege8	\|- ( ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) ) -> ( if- ( ph , ps , th ) -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) ) )
3	1 2	ax-mp	\|- ( if- ( ph , ps , th ) -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) )

Step

Hyp

Ref

Expression

frege58acor

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

ax-frege8

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

1 2

ax-mp

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