Theorem ifpxorcor

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

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

Proof

Step	Hyp	Ref	Expression
1		ifpbicor	$$⊢if-\left(\phi ,\neg \psi ,\neg \neg \psi \right)\leftrightarrow if-\left(\neg \psi ,\phi ,\neg \phi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3		ifpbi3	$$⊢\left(\psi \leftrightarrow \neg \neg \psi \right)\to \left(if-\left(\phi ,\neg \psi ,\psi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\neg \neg \psi \right)\right)$$
4	2 3	ax-mp	$$⊢if-\left(\phi ,\neg \psi ,\psi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\neg \neg \psi \right)$$
5		ifpn	$$⊢if-\left(\psi ,\neg \phi ,\phi \right)\leftrightarrow if-\left(\neg \psi ,\phi ,\neg \phi \right)$$
6	1 4 5	3bitr4i	$$⊢if-\left(\phi ,\neg \psi ,\psi \right)\leftrightarrow if-\left(\psi ,\neg \phi ,\phi \right)$$