Theorem subneintr2d

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subaddd.3	$$⊢\phi \to C\in ℂ$$
	subneintr2d.4	$$⊢\phi \to A\ne B$$
Assertion	subneintr2d	$$⊢\phi \to A-C\ne B-C$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subaddd.3	$$⊢\phi \to C\in ℂ$$
4		subneintr2d.4	$$⊢\phi \to A\ne B$$
5	1 2 3	subcan2ad	$$⊢\phi \to \left(A-C=B-C\leftrightarrow A=B\right)$$
6	5	necon3bid	$$⊢\phi \to \left(A-C\ne B-C\leftrightarrow A\ne B\right)$$
7	4 6	mpbird	$$⊢\phi \to A-C\ne B-C$$