Theorem neeq12d

Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypotheses	neeq1d.1	$$⊢\phi \to A=B$$
	neeq12d.2	$$⊢\phi \to C=D$$
Assertion	neeq12d	$$⊢\phi \to \left(A\ne C\leftrightarrow B\ne D\right)$$

Proof

Step	Hyp	Ref	Expression
1		neeq1d.1	$$⊢\phi \to A=B$$
2		neeq12d.2	$$⊢\phi \to C=D$$
3	1 2	eqeq12d	$$⊢\phi \to \left(A=C\leftrightarrow B=D\right)$$
4	3	necon3bid	$$⊢\phi \to \left(A\ne C\leftrightarrow B\ne D\right)$$