Theorem neeqtrd

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

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

Proof

Step	Hyp	Ref	Expression
1		neeqtrd.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2		neeqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	neeq2d	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )