Metamath Proof Explorer

Theorem frege9

Description: Closed form of syl with swapped antecedents. This proposition differs from frege5 only in an unessential way. Identical to imim1 . Proposition 9 of Frege1879 p. 35. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege9	$⊢ (φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		frege5	$⊢ (ψ \to χ) \to ((φ \to ψ) \to (φ \to χ))$
2		ax-frege8	$⊢ ((ψ \to χ) \to ((φ \to ψ) \to (φ \to χ))) \to ((φ \to ψ) \to ((ψ \to χ) \to (φ \to χ)))$
3	1 2	ax-mp	$⊢ (φ \to ψ) \to ((ψ \to χ) \to (φ \to χ))$