Theorem pm5.21ni

Description: Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996) (Proof shortened by Wolf Lammen, 19-May-2013)

	Ref	Expression
Hypotheses	pm5.21ni.1	$$⊢\phi \to \psi$$
	pm5.21ni.2	$$⊢\chi \to \psi$$
Assertion	pm5.21ni	$$⊢\neg \psi \to \left(\phi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.21ni.1	$$⊢\phi \to \psi$$
2		pm5.21ni.2	$$⊢\chi \to \psi$$
3	1	con3i	$$⊢\neg \psi \to \neg \phi$$
4	2	con3i	$$⊢\neg \psi \to \neg \chi$$
5	3 4	2falsed	$$⊢\neg \psi \to \left(\phi \leftrightarrow \chi \right)$$