Theorem subne0ad

Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d . Contrapositive of subeq0bd . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subne0ad.3	$$⊢\phi \to A-B\ne 0$$
Assertion	subne0ad	$$⊢\phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subne0ad.3	$$⊢\phi \to A-B\ne 0$$
4	1 2	subeq0ad	$$⊢\phi \to \left(A-B=0\leftrightarrow A=B\right)$$
5	4	necon3bid	$$⊢\phi \to \left(A-B\ne 0\leftrightarrow A\ne B\right)$$
6	3 5	mpbid	$$⊢\phi \to A\ne B$$