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Metamath Proof Explorer

Theorem necon3bd

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Hypothesis necon3bd.1 ⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
Assertion necon3bd ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 necon3bd.1 ⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝜓 ) )
2 nne ⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
3 2 1 biimtrid ⊢ ( 𝜑 → ( ¬ 𝐴 ≠ 𝐵 → 𝜓 ) )
4 3 con1d ⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 ≠ 𝐵 ) )