Theorem eqnetrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eqnetrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqnetrd.2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
3	1	neeq1d	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )
4	2 3	mpbird	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )