Theorem subcanad

Description: Cancellation law for subtraction. Deduction form of subcan . Generalization 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	$$⊢\phi \to C\in ℂ$$
Assertion	subcanad	$$⊢\phi \to \left(A-B=A-C\leftrightarrow B=C\right)$$

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		subcan	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge C\in ℂ\right)\to \left(A-B=A-C\leftrightarrow B=C\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A-B=A-C\leftrightarrow B=C\right)$$