Theorem frege65a

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a general context. Proposition 65 of Frege1879 p. 53. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege65a	|- ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpimim	|- ( if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , ch , et ) ) )
2		frege64a	|- ( ( if- ( ph , ps , ta ) -> if- ( ph , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )
3	1 2	syl	|- ( if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )
4		frege61a	|- ( ( if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) -> ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) )
5	3 4	ax-mp	|- ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )