Metamath Proof Explorer

Theorem frege67a

Description: Lemma for frege68a . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege67a	$⊢ (((ψ \land χ) \leftrightarrow θ) \to (θ \to (ψ \land χ))) \to (((ψ \land χ) \leftrightarrow θ) \to (θ \to if- (φ, ψ, χ)))$

Step	Hyp	Ref	Expression
1		ax-frege58a	$⊢ (ψ \land χ) \to if- (φ, ψ, χ)$
2		frege7	$⊢ ((ψ \land χ) \to if- (φ, ψ, χ)) \to ((((ψ \land χ) \leftrightarrow θ) \to (θ \to (ψ \land χ))) \to (((ψ \land χ) \leftrightarrow θ) \to (θ \to if- (φ, ψ, χ))))$
3	1 2	ax-mp	$⊢ (((ψ \land χ) \leftrightarrow θ) \to (θ \to (ψ \land χ))) \to (((ψ \land χ) \leftrightarrow θ) \to (θ \to if- (φ, ψ, χ)))$