Theorem frege19

Description: A closed form of syl6 . Proposition 19 of Frege1879 p. 39. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege19	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\chi \to \theta \right)\to \left(\phi \to \left(\psi \to \theta \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege9	$$⊢\left(\psi \to \chi \right)\to \left(\left(\chi \to \theta \right)\to \left(\psi \to \theta \right)\right)$$
2		frege18	$$⊢\left(\left(\psi \to \chi \right)\to \left(\left(\chi \to \theta \right)\to \left(\psi \to \theta \right)\right)\right)\to \left(\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\chi \to \theta \right)\to \left(\phi \to \left(\psi \to \theta \right)\right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\chi \to \theta \right)\to \left(\phi \to \left(\psi \to \theta \right)\right)\right)$$