Theorem addneintr2d

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad . Consequence of addcan2d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	muld.1	$$⊢\phi \to A\in ℂ$$
	addcomd.2	$$⊢\phi \to B\in ℂ$$
	addcand.3	$$⊢\phi \to C\in ℂ$$
	addneintr2d.4	$$⊢\phi \to A\ne B$$
Assertion	addneintr2d	$$⊢\phi \to A+C\ne B+C$$

Proof

Step	Hyp	Ref	Expression
1		muld.1	$$⊢\phi \to A\in ℂ$$
2		addcomd.2	$$⊢\phi \to B\in ℂ$$
3		addcand.3	$$⊢\phi \to C\in ℂ$$
4		addneintr2d.4	$$⊢\phi \to A\ne B$$
5	1 2 3	addcan2d	$$⊢\phi \to \left(A+C=B+C\leftrightarrow A=B\right)$$
6	5	necon3bid	$$⊢\phi \to \left(A+C\ne B+C\leftrightarrow A\ne B\right)$$
7	4 6	mpbird	$$⊢\phi \to A+C\ne B+C$$