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Metamath Proof Explorer

Theorem necon2bd

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

Ref Expression
Hypothesis necon2bd.1 ⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )
Assertion necon2bd ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 necon2bd.1 ⊢ ( 𝜑 → ( 𝜓 → 𝐴 ≠ 𝐵 ) )
2 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3 1 2 imbitrdi ⊢ ( 𝜑 → ( 𝜓 → ¬ 𝐴 = 𝐵 ) )
4 3 con2d ⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ¬ 𝜓 ) )