Theorem ifpnot23d

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

	Ref	Expression
Assertion	ifpnot23d	$$⊢\neg if-\left(\phi ,\neg \psi ,\neg \chi \right)\leftrightarrow if-\left(\phi ,\psi ,\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ifpnot23	$$⊢\neg if-\left(\phi ,\neg \psi ,\neg \chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \neg \chi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3		notnotb	$$⊢\chi \leftrightarrow \neg \neg \chi$$
4		ifpbi23	$$⊢\left(\left(\psi \leftrightarrow \neg \neg \psi \right)\wedge \left(\chi \leftrightarrow \neg \neg \chi \right)\right)\to \left(if-\left(\phi ,\psi ,\chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \neg \chi \right)\right)$$
5	2 3 4	mp2an	$$⊢if-\left(\phi ,\psi ,\chi \right)\leftrightarrow if-\left(\phi ,\neg \neg \psi ,\neg \neg \chi \right)$$
6	1 5	bitr4i	$$⊢\neg if-\left(\phi ,\neg \psi ,\neg \chi \right)\leftrightarrow if-\left(\phi ,\psi ,\chi \right)$$