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Metamath Proof Explorer

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 ⊢ ( 𝜑 → ¬ 𝜓 )
2falsed.2 ⊢ ( 𝜑 → ¬ 𝜒 )
Assertion 2falsed ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 2falsed.1 ⊢ ( 𝜑 → ¬ 𝜓 )
2 2falsed.2 ⊢ ( 𝜑 → ¬ 𝜒 )
3 1 2 2thd ⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
4 3 con4bid ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )