Theorem subneintrd

Description: Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand . (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	$$⊢\phi \to C\in ℂ$$
	subneintrd.4	$$⊢\phi \to B\ne C$$
Assertion	subneintrd	$$⊢\phi \to A-B\ne A-C$$

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		subneintrd.4	$$⊢\phi \to B\ne C$$
5	1 2 3	subcanad	$$⊢\phi \to \left(A-B=A-C\leftrightarrow B=C\right)$$
6	5	necon3bid	$$⊢\phi \to \left(A-B\ne A-C\leftrightarrow B\ne C\right)$$
7	4 6	mpbird	$$⊢\phi \to A-B\ne A-C$$