Theorem impor 38029

Description: An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)

Assertion

Ref	Expression
impor	⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒))

Proof of Theorem impor

Step	Hyp	Ref	Expression
1		imor 853	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (¬ 𝜑 ∨ (𝜓 ∨ 𝜒)))
2		orass 921	. 2 ⊢ (((¬ 𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ 𝜑 ∨ (𝜓 ∨ 𝜒)))
3	1, 2	bitr4i 278	1 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847

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 207  df-or 848

This theorem is referenced by: (None)