This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem necon3i

Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006) (Proof shortened by Wolf Lammen, 22-Nov-2019)

Ref Expression
Hypothesis necon3i.1 ⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐷 )
Assertion necon3i ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 )

Proof

Step Hyp Ref Expression
1 necon3i.1 ⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐷 )
2 1 necon3ai ⊢ ( 𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵 )
3 2 neqned ⊢ ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 )