Metamath Proof Explorer

Theorem frege6

Description: A closed form of imim2d which is a deduction adding nested antecedents. Proposition 6 of Frege1879 p. 33. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege6	$⊢ (φ \to (ψ \to χ)) \to (φ \to ((θ \to ψ) \to (θ \to χ)))$

Proof

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