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 ) )