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Metamath Proof Explorer

Theorem necon4d

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

Ref Expression
Hypothesis necon4d.1 ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 ) )
Assertion necon4d ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 necon4d.1 ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷 ) )
2 1 necon2bd ⊢ ( 𝜑 → ( 𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵 ) )
3 nne ⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 )
4 2 3 imbitrdi ⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝐴 = 𝐵 ) )