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Metamath Proof Explorer

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 ) )