Theorem frege37

Description: If ch is a necessary consequence of the occurrence of ps or ph , then ch is a necessary consequence of ph alone. Similar to a closed form of orcs . Proposition 37 of Frege1879 p. 46. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege37	$$⊢\left(\left(\neg \phi \to \psi \right)\to \chi \right)\to \left(\phi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		frege36	$$⊢\phi \to \left(\neg \phi \to \psi \right)$$
2		frege9	$$⊢\left(\phi \to \left(\neg \phi \to \psi \right)\right)\to \left(\left(\left(\neg \phi \to \psi \right)\to \chi \right)\to \left(\phi \to \chi \right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left(\neg \phi \to \psi \right)\to \chi \right)\to \left(\phi \to \chi \right)$$