Theorem diveq1ad

Description: The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 . Generalization of diveq1d . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divcld.3	$$⊢\phi \to B\ne 0$$
Assertion	diveq1ad	$$⊢\phi \to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divcld.3	$$⊢\phi \to B\ne 0$$
4		diveq1	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(\frac{A}{B}=1\leftrightarrow A=B\right)$$