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Metamath Proof Explorer

Theorem necon1d

Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Hypothesis necon1d.1 ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 ) )
Assertion necon1d ( 𝜑 → ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 necon1d.1 ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 ) )
2 nne ⊢ ( ¬ 𝐶 ≠ 𝐷 ↔ 𝐶 = 𝐷 )
3 1 2 imbitrrdi ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → ¬ 𝐶 ≠ 𝐷 ) )
4 3 necon4ad ⊢ ( 𝜑 → ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 ) )