Theorem div11d

Description: One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divmuld.3	$$⊢\phi \to C\in ℂ$$
	divassd.4	$$⊢\phi \to C\ne 0$$
	div11d.5	$$⊢\phi \to \frac{A}{C}=\frac{B}{C}$$
Assertion	div11d	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divmuld.3	$$⊢\phi \to C\in ℂ$$
4		divassd.4	$$⊢\phi \to C\ne 0$$
5		div11d.5	$$⊢\phi \to \frac{A}{C}=\frac{B}{C}$$
6		div11	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \left(\frac{A}{C}=\frac{B}{C}\leftrightarrow A=B\right)$$
7	1 2 3 4 6	syl112anc	$$⊢\phi \to \left(\frac{A}{C}=\frac{B}{C}\leftrightarrow A=B\right)$$
8	5 7	mpbid	$$⊢\phi \to A=B$$