Theorem divdiv32i

Description: Swap denominators in a division. (Contributed by NM, 15-Sep-1999)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divmulz.3	$$⊢C\in ℂ$$
	divmul.4	$$⊢B\ne 0$$
	divdiv23.5	$$⊢C\ne 0$$
Assertion	divdiv32i	$$⊢\frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divmul.4	$$⊢B\ne 0$$
5		divdiv23.5	$$⊢C\ne 0$$
6	1 2 3	divdiv23zi	$$⊢\left(B\ne 0\wedge C\ne 0\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$
7	4 5 6	mp2an	$$⊢\frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$