Theorem divdiv23zi

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
Assertion	divdiv23zi	⊢ ( ( 𝐵 ≠ 0 ∧ 𝐶 ≠ 0 ) → ( ( 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		divdiv32	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) )
5	1 4	mp3an1	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) )
6	3 5	mpanr1	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) )
7	2 6	mpanl1	⊢ ( ( 𝐵 ≠ 0 ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) )