Theorem necon1ai

Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007) (Proof shortened by Wolf Lammen, 22-Nov-2019)

	Ref	Expression
Hypothesis	necon1ai.1	⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 )
Assertion	necon1ai	⊢ ( 𝐴 ≠ 𝐵 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		necon1ai.1	⊢ ( ¬ 𝜑 → 𝐴 = 𝐵 )
2	1	necon3ai	⊢ ( 𝐴 ≠ 𝐵 → ¬ ¬ 𝜑 )
3	2	notnotrd	⊢ ( 𝐴 ≠ 𝐵 → 𝜑 )