Theorem ifpor

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

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

Proof

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