Theorem frege18

Description: Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of Frege1879 p. 39. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege18	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege5	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜃 → ( 𝜓 → 𝜒 ) ) ) )
2		frege16	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜃 → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )