Theorem ifpnot23b

Description: Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpnot23b	$$⊢\neg if-\left(\phi ,\neg \psi ,\chi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ifpnot23	$$⊢\neg if-\left(\phi ,\neg \psi ,\chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \chi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3		ifpbi2	$$⊢\left(\psi \leftrightarrow \neg \neg \psi \right)\to \left(if-\left(\phi ,\psi ,\neg \chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \chi \right)\right)$$
4	2 3	ax-mp	$$⊢if-\left(\phi ,\psi ,\neg \chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \chi \right)$$
5	1 4	bitr4i	$$⊢\neg if-\left(\phi ,\neg \psi ,\chi \right)\leftrightarrow if-\left(\phi ,\psi ,\neg \chi \right)$$