Theorem ifpn

Description: Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019) (Proof shortened by Wolf Lammen, 5-May-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(\left(\neg \phi \vee \psi \right)\wedge \left(\neg \phi \to \chi \right)\right)\leftrightarrow \left(\left(\neg \phi \to \chi \right)\wedge \left(\neg \phi \vee \psi \right)\right)$$
2		dfifp5	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow \left(\left(\neg \phi \vee \psi \right)\wedge \left(\neg \phi \to \chi \right)\right)$$
3		dfifp3	$$⊢if-\left(\neg \phi ,\chi ,\psi \right)\leftrightarrow \left(\left(\neg \phi \to \chi \right)\wedge \left(\neg \phi \vee \psi \right)\right)$$
4	1 2 3	3bitr4i	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow if-\left(\neg \phi ,\chi ,\psi \right)$$