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Metamath Proof Explorer

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