Theorem bj-bijust00

Description: A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust is an instance ( bijust0 and bj-bijust0ALT are therefore also instances of it). (Contributed by BJ, 7-Sep-2022)

	Ref	Expression
Assertion	bj-bijust00	$$⊢\neg \left(\left(\phi \to \phi \right)\to \neg \left(\psi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\phi \to \phi$$
2		id	$$⊢\psi \to \psi$$
3		pm3.2im	$$⊢\left(\phi \to \phi \right)\to \left(\left(\psi \to \psi \right)\to \neg \left(\left(\phi \to \phi \right)\to \neg \left(\psi \to \psi \right)\right)\right)$$
4	1 2 3	mp2	$$⊢\neg \left(\left(\phi \to \phi \right)\to \neg \left(\psi \to \psi \right)\right)$$