Theorem necon3d

Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006)

	Ref	Expression
Hypothesis	necon3d.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝐶 = 𝐷 ) )
Assertion	necon3d	⊢ ( 𝜑 → ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		necon3d.1	⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝐶 = 𝐷 ) )
2	1	necon3ad	⊢ ( 𝜑 → ( 𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵 ) )
3		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
4	2 3	syl6ibr	⊢ ( 𝜑 → ( 𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵 ) )