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Metamath Proof Explorer

Theorem necon1i

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007)

Ref Expression
Hypothesis necon1i.1 ⊢ ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 )
Assertion necon1i ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 necon1i.1 ⊢ ( 𝐴 ≠ 𝐵 → 𝐶 = 𝐷 )
2 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3 2 1 sylbir ⊢ ( ¬ 𝐴 = 𝐵 → 𝐶 = 𝐷 )
4 3 necon1ai ⊢ ( 𝐶 ≠ 𝐷 → 𝐴 = 𝐵 )