Theorem bj-orim2

Description: Proof of orim2 from the axiomatic definition of disjunction ( olc , orc , jao ) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-orim2	$$⊢\left(\phi \to \psi \right)\to \left(\left(\chi \vee \phi \right)\to \left(\chi \vee \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orc	$$⊢\chi \to \left(\chi \vee \psi \right)$$
2		olc	$$⊢\psi \to \left(\chi \vee \psi \right)$$
3	2	imim2i	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \left(\chi \vee \psi \right)\right)$$
4		jao	$$⊢\left(\chi \to \left(\chi \vee \psi \right)\right)\to \left(\left(\phi \to \left(\chi \vee \psi \right)\right)\to \left(\left(\chi \vee \phi \right)\to \left(\chi \vee \psi \right)\right)\right)$$
5	1 3 4	mpsyl	$$⊢\left(\phi \to \psi \right)\to \left(\left(\chi \vee \phi \right)\to \left(\chi \vee \psi \right)\right)$$