Theorem orimdi

Description: Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013)

	Ref	Expression
Assertion	orimdi	⊢ ( ( 𝜑 ∨ ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imdi	⊢ ( ( ¬ 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜑 → 𝜒 ) ) )
2		df-or	⊢ ( ( 𝜑 ∨ ( 𝜓 → 𝜒 ) ) ↔ ( ¬ 𝜑 → ( 𝜓 → 𝜒 ) ) )
3		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
4		df-or	⊢ ( ( 𝜑 ∨ 𝜒 ) ↔ ( ¬ 𝜑 → 𝜒 ) )
5	3 4	imbi12i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ 𝜒 ) ) ↔ ( ( ¬ 𝜑 → 𝜓 ) → ( ¬ 𝜑 → 𝜒 ) ) )
6	1 2 5	3bitr4i	⊢ ( ( 𝜑 ∨ ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ 𝜒 ) ) )