Theorem divcan3zi

Description: A cancellation law for division. (Eliminates a hypothesis of divcan3i with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
Assertion	divcan3zi	$$⊢B\ne 0\to \frac{BA}{B}=A$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divcan3	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{BA}{B}=A$$
4	1 2 3	mp3an12	$$⊢B\ne 0\to \frac{BA}{B}=A$$