Theorem elif

Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005)

	Ref	Expression
Assertion	elif	$$⊢A\in if\left(\phi ,B,C\right)\leftrightarrow \left(\left(\phi \wedge A\in B\right)\vee \left(\neg \phi \wedge A\in C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢if\left(\phi ,B,C\right)=B\to \left(A\in if\left(\phi ,B,C\right)\leftrightarrow A\in B\right)$$
2		eleq2	$$⊢if\left(\phi ,B,C\right)=C\to \left(A\in if\left(\phi ,B,C\right)\leftrightarrow A\in C\right)$$
3	1 2	elimif	$$⊢A\in if\left(\phi ,B,C\right)\leftrightarrow \left(\left(\phi \wedge A\in B\right)\vee \left(\neg \phi \wedge A\in C\right)\right)$$