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Metamath Proof Explorer

Theorem pncan2

Description: Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005)

		Ref	Expression
	Assertion	pncan2	$⊢ (A \in ℂ \land B \in ℂ) \to A + B - A = B$

Step	Hyp	Ref	Expression
1		addcom	$⊢ (B \in ℂ \land A \in ℂ) \to B + A = A + B$
2	1	oveq1d	$⊢ (B \in ℂ \land A \in ℂ) \to B + A - A = A + B - A$
3		pncan	$⊢ (B \in ℂ \land A \in ℂ) \to B + A - A = B$
4	2 3	eqtr3d	$⊢ (B \in ℂ \land A \in ℂ) \to A + B - A = B$
5	4	ancoms	$⊢ (A \in ℂ \land B \in ℂ) \to A + B - A = B$