Theorem 2falsed

Description: Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013) (Proof shortened by Wolf Lammen, 11-Apr-2024)

	Ref	Expression
Hypotheses	2falsed.1	$$⊢\phi \to \neg \psi$$
	2falsed.2	$$⊢\phi \to \neg \chi$$
Assertion	2falsed	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2falsed.1	$$⊢\phi \to \neg \psi$$
2		2falsed.2	$$⊢\phi \to \neg \chi$$
3	1 2	2thd	$$⊢\phi \to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
4	3	con4bid	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$