Theorem subeq0ad

Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 . Generalization of subeq0d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
Assertion	subeq0ad	$$⊢\phi \to \left(A-B=0\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subeq0	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A-B=0\leftrightarrow A=B\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A-B=0\leftrightarrow A=B\right)$$