Theorem ifpim3

Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpim3	$$⊢\left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
2		orc	$$⊢\phi \to \left(\phi \vee \psi \right)$$
3		ifpim23g	$$⊢\left(\left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \phi \right)\right)\leftrightarrow \left(\left(\left(\phi \wedge \psi \right)\to \phi \right)\wedge \left(\phi \to \left(\phi \vee \psi \right)\right)\right)$$
4	1 2 3	mpbir2an	$$⊢\left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \phi \right)$$