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	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ∨ 𝜑 ) → ( 𝜒 ∨ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝜒 → ( 𝜒 ∨ 𝜓 ) )
2		olc	⊢ ( 𝜓 → ( 𝜒 ∨ 𝜓 ) )
3	2	imim2i	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜒 ∨ 𝜓 ) ) )
4		jao	⊢ ( ( 𝜒 → ( 𝜒 ∨ 𝜓 ) ) → ( ( 𝜑 → ( 𝜒 ∨ 𝜓 ) ) → ( ( 𝜒 ∨ 𝜑 ) → ( 𝜒 ∨ 𝜓 ) ) ) )
5	1 3 4	mpsyl	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ∨ 𝜑 ) → ( 𝜒 ∨ 𝜓 ) ) )