Theorem neeq1d

Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999) (Proof shortened by Wolf Lammen, 19-Nov-2019)

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

Proof

Step	Hyp	Ref	Expression
1		neeq1d.1	$$⊢\phi \to A=B$$
2	1	eqeq1d	$$⊢\phi \to \left(A=C\leftrightarrow B=C\right)$$
3	2	necon3bid	$$⊢\phi \to \left(A\ne C\leftrightarrow B\ne C\right)$$