Theorem eqnetrd

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

	Ref	Expression
Hypotheses	eqnetrd.1	$$⊢\phi \to A=B$$
	eqnetrd.2	$$⊢\phi \to B\ne C$$
Assertion	eqnetrd	$$⊢\phi \to A\ne C$$

Proof

Step	Hyp	Ref	Expression
1		eqnetrd.1	$$⊢\phi \to A=B$$
2		eqnetrd.2	$$⊢\phi \to B\ne C$$
3	1	neeq1d	$$⊢\phi \to \left(A\ne C\leftrightarrow B\ne C\right)$$
4	2 3	mpbird	$$⊢\phi \to A\ne C$$