Theorem ifpim2

Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020)

	Ref	Expression
Assertion	ifpim2	$$⊢\left(\phi \to \psi \right)\leftrightarrow if-\left(\psi ,\top ,\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		tru	$$⊢\top$$
2	1	olci	$$⊢\left(\neg \psi \vee \top \right)$$
3	2	biantrur	$$⊢\left(\psi \vee \neg \phi \right)\leftrightarrow \left(\left(\neg \psi \vee \top \right)\wedge \left(\psi \vee \neg \phi \right)\right)$$
4		imor	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\neg \phi \vee \psi \right)$$
5		orcom	$$⊢\left(\neg \phi \vee \psi \right)\leftrightarrow \left(\psi \vee \neg \phi \right)$$
6	4 5	bitri	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\psi \vee \neg \phi \right)$$
7		dfifp4	$$⊢if-\left(\psi ,\top ,\neg \phi \right)\leftrightarrow \left(\left(\neg \psi \vee \top \right)\wedge \left(\psi \vee \neg \phi \right)\right)$$
8	3 6 7	3bitr4i	$$⊢\left(\phi \to \psi \right)\leftrightarrow if-\left(\psi ,\top ,\neg \phi \right)$$