Metamath Proof Explorer

Theorem frege56a

Description: Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege56a	$⊢ ((φ \leftrightarrow ψ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ))) \to ((ψ \leftrightarrow φ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ)))$

Step	Hyp	Ref	Expression
1		frege55cor1a	$⊢ (ψ \leftrightarrow φ) \to (φ \leftrightarrow ψ)$
2		frege9	$⊢ ((ψ \leftrightarrow φ) \to (φ \leftrightarrow ψ)) \to (((φ \leftrightarrow ψ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ))) \to ((ψ \leftrightarrow φ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ))))$
3	1 2	ax-mp	$⊢ ((φ \leftrightarrow ψ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ))) \to ((ψ \leftrightarrow φ) \to (if- (φ, χ, θ) \to if- (ψ, χ, θ)))$