Theorem ifel

Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005)

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

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢if\left(\phi ,A,B\right)=A\to \left(if\left(\phi ,A,B\right)\in C\leftrightarrow A\in C\right)$$
2		eleq1	$$⊢if\left(\phi ,A,B\right)=B\to \left(if\left(\phi ,A,B\right)\in C\leftrightarrow B\in C\right)$$
3	1 2	elimif	$$⊢if\left(\phi ,A,B\right)\in C\leftrightarrow \left(\left(\phi \wedge A\in C\right)\vee \left(\neg \phi \wedge B\in C\right)\right)$$