Theorem neg11d

Description: If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	neg11d.3	$$⊢\phi \to -A=-B$$
Assertion	neg11d	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		neg11d.3	$$⊢\phi \to -A=-B$$
4	1 2	neg11ad	$$⊢\phi \to \left(-A=-B\leftrightarrow A=B\right)$$
5	3 4	mpbid	$$⊢\phi \to A=B$$