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Metamath Proof Explorer

Theorem divcan3zi

Description: A cancellation law for division. (Eliminates a hypothesis of divcan3i with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004)

	Ref	Expression
Hypotheses	divclz.1	$⊢ A \in ℂ$
	divclz.2	$⊢ B \in ℂ$
Assertion	divcan3zi	$⊢ B \neq 0 \to \frac{B A}{B} = A$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$⊢ A \in ℂ$
2		divclz.2	$⊢ B \in ℂ$
3		divcan3	$⊢ (A \in ℂ \land B \in ℂ \land B \neq 0) \to \frac{B A}{B} = A$
4	1 2 3	mp3an12	$⊢ B \neq 0 \to \frac{B A}{B} = A$