Theorem ifpnim1

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

	Ref	Expression
Assertion	ifpnim1	$$⊢\neg \left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ifpnot23c	$$⊢\neg if-\left(\phi ,\psi ,\neg \phi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\phi \right)$$
2		ifpim3	$$⊢\left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \phi \right)$$
3	1 2	xchnxbir	$$⊢\neg \left(\phi \to \psi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\phi \right)$$