Theorem eqnetrrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eqnetrrd.1	$$⊢\phi \to A=B$$
2		eqnetrrd.2	$$⊢\phi \to A\ne C$$
3	1	eqcomd	$$⊢\phi \to B=A$$
4	3 2	eqnetrd	$$⊢\phi \to B\ne C$$