Theorem subeq0d

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	$$⊢\phi \to B\in ℂ$$
	subeq0d.3	$$⊢\phi \to A-B=0$$
Assertion	subeq0d	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subeq0d.3	$$⊢\phi \to A-B=0$$
4		subeq0	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(A-B=0\leftrightarrow A=B\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A-B=0\leftrightarrow A=B\right)$$
6	3 5	mpbid	$$⊢\phi \to A=B$$