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	$$⊢if-\left(\phi ,\psi ,\chi \right)\to \left(\psi \vee \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ifp	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \wedge \psi \right)\vee \left(\neg \phi \wedge \chi \right)\right)$$
2		simpr	$$⊢\left(\phi \wedge \psi \right)\to \psi$$
3		simpr	$$⊢\left(\neg \phi \wedge \chi \right)\to \chi$$
4	2 3	orim12i	$$⊢\left(\left(\phi \wedge \psi \right)\vee \left(\neg \phi \wedge \chi \right)\right)\to \left(\psi \vee \chi \right)$$
5	1 4	sylbi	$$⊢if-\left(\phi ,\psi ,\chi \right)\to \left(\psi \vee \chi \right)$$