Theorem neeqtrrd

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

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

Proof

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