Theorem divdiv23zi

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	$$⊢C\in ℂ$$
Assertion	divdiv23zi	$$⊢\left(B\ne 0\wedge C\ne 0\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divdiv32	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$
5	1 4	mp3an1	$$⊢\left(\left(B\in ℂ\wedge B\ne 0\right)\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$
6	3 5	mpanr1	$$⊢\left(\left(B\in ℂ\wedge B\ne 0\right)\wedge C\ne 0\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$
7	2 6	mpanl1	$$⊢\left(B\ne 0\wedge C\ne 0\right)\to \frac{\frac{A}{B}}{C}=\frac{\frac{A}{C}}{B}$$