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	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpimim	⊢ ( if- ( 𝜑 , ( 𝜓 → 𝜒 ) , ( 𝜏 → 𝜂 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜒 , 𝜂 ) ) )
2		frege64a	⊢ ( ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜒 , 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )
3	1 2	syl	⊢ ( if- ( 𝜑 , ( 𝜓 → 𝜒 ) , ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )
4		frege61a	⊢ ( ( if- ( 𝜑 , ( 𝜓 → 𝜒 ) , ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )