Metamath Proof Explorer

Theorem frege3

Description: Add antecedent to ax-frege2 . Special case of rp-frege3g . Proposition 3 of Frege1879 p. 29. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege3	$⊢ (φ \to ψ) \to ((χ \to (φ \to ψ)) \to ((χ \to φ) \to (χ \to ψ)))$

Proof

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