Theorem bj-orim2 34736

Description: Proof of orim2 965 from the axiomatic definition of disjunction (olc 865, orc 864, jao 958) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-orim2	⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))

Proof of Theorem bj-orim2

Step	Hyp	Ref	Expression
1		orc 864	. 2 ⊢ (𝜒 → (𝜒 ∨ 𝜓))
2		olc 865	. . 3 ⊢ (𝜓 → (𝜒 ∨ 𝜓))
3	2	imim2i 16	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 ∨ 𝜓)))
4		jao 958	. 2 ⊢ ((𝜒 → (𝜒 ∨ 𝜓)) → ((𝜑 → (𝜒 ∨ 𝜓)) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓))))
5	1, 3, 4	mpsyl 68	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))

Syntax hints:   → wi 4   ∨ wo 844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845

This theorem is referenced by: (None)