Theorem clifte

Description: show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020)

	Ref	Expression
Hypotheses	clifte.1	$$⊢\left(\phi \wedge \neg \chi \right)$$
	clifte.2	$$⊢\theta$$
Assertion	clifte	$$⊢\theta \leftrightarrow \left(\left(\phi \wedge \neg \chi \right)\vee \left(\psi \wedge \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		clifte.1	$$⊢\left(\phi \wedge \neg \chi \right)$$
2		clifte.2	$$⊢\theta$$
3	1	orci	$$⊢\left(\left(\phi \wedge \neg \chi \right)\vee \left(\psi \wedge \chi \right)\right)$$
4	2 3	2th	$$⊢\theta \leftrightarrow \left(\left(\phi \wedge \neg \chi \right)\vee \left(\psi \wedge \chi \right)\right)$$