Theorem divdiv32i

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

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divmulz.3	⊢ 𝐶 ∈ ℂ
	divmul.4	⊢ 𝐵 ≠ 0
	divdiv23.5	⊢ 𝐶 ≠ 0
Assertion	divdiv32i	⊢ ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divmulz.3	⊢ 𝐶 ∈ ℂ
4		divmul.4	⊢ 𝐵 ≠ 0
5		divdiv23.5	⊢ 𝐶 ≠ 0
6	1 2 3	divdiv23zi	⊢ ( ( 𝐵 ≠ 0 ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) )
7	4 5 6	mp2an	⊢ ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 )