Theorem eqnetrrd

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

	Ref	Expression
Hypotheses	eqnetrrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eqnetrrd.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
Assertion	eqnetrrd	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqnetrrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqnetrrd.2	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )
3	1	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
4	3 2	eqnetrd	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )