Theorem necon1ad

Description: Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypothesis	necon1ad.1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) )
Assertion	necon1ad	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		necon1ad.1	⊢ ( 𝜑 → ( ¬ 𝜓 → 𝐴 = 𝐵 ) )
2	1	necon3ad	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → ¬ ¬ 𝜓 ) )
3		notnotr	⊢ ( ¬ ¬ 𝜓 → 𝜓 )
4	2 3	syl6	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 → 𝜓 ) )