Theorem bj-orim2 32541

Description: Proof of orim2 886 from the axiomatic definition of disjunction (olc 399, orc 400, jao 534) 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 400	. 2 ⊢ (𝜒 → (𝜒 ∨ 𝜓))
2		olc 399	. . 3 ⊢ (𝜓 → (𝜒 ∨ 𝜓))
3	2	imim2i 16	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 ∨ 𝜓)))
4		jao 534	. 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 383

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 197  df-or 385  df-an 386

This theorem is referenced by:  bj-peirce  32543