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	⊢ ( 𝜑 → 𝜓 )
	pm5.21ni.2	⊢ ( 𝜒 → 𝜓 )
Assertion	pm5.21ni	⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.21ni.1	⊢ ( 𝜑 → 𝜓 )
2		pm5.21ni.2	⊢ ( 𝜒 → 𝜓 )
3	1	con3i	⊢ ( ¬ 𝜓 → ¬ 𝜑 )
4	2	con3i	⊢ ( ¬ 𝜓 → ¬ 𝜒 )
5	3 4	2falsed	⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) )