Metamath Proof Explorer

Theorem rp-frege3g

Description: Add antecedent to ax-frege2 . More general statement than frege3 . Like ax-frege2 , it is essentially a closed form of mpd , however it has an extra antecedent.

It would be more natural to prove from a1i and ax-frege2 in Metamath. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	rp-frege3g	$⊢ φ \to ((ψ \to (χ \to θ)) \to ((ψ \to χ) \to (ψ \to θ)))$

Proof

Step	Hyp	Ref	Expression
1		ax-frege2	$⊢ (ψ \to (χ \to θ)) \to ((ψ \to χ) \to (ψ \to θ))$
2		ax-frege1	$⊢ ((ψ \to (χ \to θ)) \to ((ψ \to χ) \to (ψ \to θ))) \to (φ \to ((ψ \to (χ \to θ)) \to ((ψ \to χ) \to (ψ \to θ))))$
3	1 2	ax-mp	$⊢ φ \to ((ψ \to (χ \to θ)) \to ((ψ \to χ) \to (ψ \to θ)))$