Theorem nemtbir

Description: An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007)

	Ref	Expression
Hypotheses	nemtbir.1	⊢ 𝐴 ≠ 𝐵
	nemtbir.2	⊢ ( 𝜑 ↔ 𝐴 = 𝐵 )
Assertion	nemtbir	⊢ ¬ 𝜑

Proof

Step	Hyp	Ref	Expression
1		nemtbir.1	⊢ 𝐴 ≠ 𝐵
2		nemtbir.2	⊢ ( 𝜑 ↔ 𝐴 = 𝐵 )
3	1	neii	⊢ ¬ 𝐴 = 𝐵
4	3 2	mtbir	⊢ ¬ 𝜑