Theorem ifpor 1072

Description: The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1071. (Contributed by BJ, 1-Oct-2019.)

Assertion

Ref	Expression
ifpor	⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒))

Proof of Theorem ifpor

Step	Hyp	Ref	Expression
1		df-ifp 1063	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
3		simpr 484	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜒)
4	2, 3	orim12i 908	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (𝜓 ∨ 𝜒))
5	1, 4	sylbi 217	1 ⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847  if-wif 1062

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-an 396  df-or 848  df-ifp 1063

This theorem is referenced by: (None)