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Metamath Proof Explorer

Theorem addsubsub23

Description: Swap the second and the third terms in a difference of a sum and a difference (or, vice versa, in a sum of a difference and a sum). (Contributed by AV, 15-Nov-2025)

		Ref	Expression
	Hypotheses	negidd.1	$⊢ φ \to A \in ℂ$
		pncand.2	$⊢ φ \to B \in ℂ$
		subaddd.3	$⊢ φ \to C \in ℂ$
		addsub4d.4	$⊢ φ \to D \in ℂ$
	Assertion	addsubsub23	$⊢ φ \to A + B - (C - D) = (A - C) + B + D$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$⊢ φ \to A \in ℂ$
2		pncand.2	$⊢ φ \to B \in ℂ$
3		subaddd.3	$⊢ φ \to C \in ℂ$
4		addsub4d.4	$⊢ φ \to D \in ℂ$
5	1 2	addcld	$⊢ φ \to A + B \in ℂ$
6	5 3 4	subsubd	$⊢ φ \to A + B - (C - D) = (A + B) - C + D$
7	1 2 3	addsubd	$⊢ φ \to A + B - C = A - C + B$
8	7	oveq1d	$⊢ φ \to (A + B) - C + D = (A - C) + B + D$
9	1 3	subcld	$⊢ φ \to A - C \in ℂ$
10	9 2 4	addassd	$⊢ φ \to (A - C) + B + D = (A - C) + B + D$
11	6 8 10	3eqtrd	$⊢ φ \to A + B - (C - D) = (A - C) + B + D$