Theorem eqnetri

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)

	Ref	Expression
Hypotheses	eqnetr.1	⊢ 𝐴 = 𝐵
	eqnetr.2	⊢ 𝐵 ≠ 𝐶
Assertion	eqnetri	⊢ 𝐴 ≠ 𝐶

Proof

Step	Hyp	Ref	Expression
1		eqnetr.1	⊢ 𝐴 = 𝐵
2		eqnetr.2	⊢ 𝐵 ≠ 𝐶
3	1	neeq1i	⊢ ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 )
4	2 3	mpbir	⊢ 𝐴 ≠ 𝐶