Theorem ifpnancor

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

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

Proof

Step	Hyp	Ref	Expression
1		ifpancor	$$⊢if-\left(\phi ,\psi ,\phi \right)\leftrightarrow if-\left(\psi ,\phi ,\psi \right)$$
2	1	notbii	$$⊢\neg if-\left(\phi ,\psi ,\phi \right)\leftrightarrow \neg if-\left(\psi ,\phi ,\psi \right)$$
3		ifpnot23	$$⊢\neg if-\left(\phi ,\psi ,\phi \right)\leftrightarrow if-\left(\phi ,\neg \psi ,\neg \phi \right)$$
4		ifpnot23	$$⊢\neg if-\left(\psi ,\phi ,\psi \right)\leftrightarrow if-\left(\psi ,\neg \phi ,\neg \psi \right)$$
5	2 3 4	3bitr3i	$$⊢if-\left(\phi ,\neg \psi ,\neg \phi \right)\leftrightarrow if-\left(\psi ,\neg \phi ,\neg \psi \right)$$