Theorem dfifp3

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019)

	Ref	Expression
Assertion	dfifp3	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\phi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfifp2	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\neg \phi \to \chi \right)\right)$$
2		pm4.64	$$⊢\left(\neg \phi \to \chi \right)\leftrightarrow \left(\phi \vee \chi \right)$$
3	2	anbi2i	$$⊢\left(\left(\phi \to \psi \right)\wedge \left(\neg \phi \to \chi \right)\right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\phi \vee \chi \right)\right)$$
4	1 3	bitri	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\phi \vee \chi \right)\right)$$