Theorem subne0d

Description: Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subne0d.3	$$⊢\phi \to A\ne B$$
Assertion	subne0d	$$⊢\phi \to A-B\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subne0d.3	$$⊢\phi \to A\ne B$$
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	5	necon3bid	$$⊢\phi \to \left(A-B\ne 0\leftrightarrow A\ne B\right)$$
7	3 6	mpbird	$$⊢\phi \to A-B\ne 0$$