Theorem 3netr4d

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 21-Nov-2019)

	Ref	Expression
Hypotheses	3netr4d.1	$$⊢\phi \to A\ne B$$
	3netr4d.2	$$⊢\phi \to C=A$$
	3netr4d.3	$$⊢\phi \to D=B$$
Assertion	3netr4d	$$⊢\phi \to C\ne D$$

Proof

Step	Hyp	Ref	Expression
1		3netr4d.1	$$⊢\phi \to A\ne B$$
2		3netr4d.2	$$⊢\phi \to C=A$$
3		3netr4d.3	$$⊢\phi \to D=B$$
4	2 1	eqnetrd	$$⊢\phi \to C\ne B$$
5	4 3	neeqtrrd	$$⊢\phi \to C\ne D$$