Theorem ifpbicor

Description: Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020)

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

Proof

Step	Hyp	Ref	Expression
1		bicom	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\psi \leftrightarrow \phi \right)$$
2		ifpdfbi	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \psi \right)$$
3		ifpdfbi	$$⊢\left(\psi \leftrightarrow \phi \right)\leftrightarrow if-\left(\psi ,\phi ,\neg \phi \right)$$
4	1 2 3	3bitr3i	$$⊢if-\left(\phi ,\psi ,\neg \psi \right)\leftrightarrow if-\left(\psi ,\phi ,\neg \phi \right)$$