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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	\|- ( ph -> A e. CC )
		pncand.2	\|- ( ph -> B e. CC )
		subaddd.3	\|- ( ph -> C e. CC )
		addsub4d.4	\|- ( ph -> D e. CC )
	Assertion	addsubsub23	\|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( A - C ) + ( B + D ) ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	\|- ( ph -> A e. CC )
2		pncand.2	\|- ( ph -> B e. CC )
3		subaddd.3	\|- ( ph -> C e. CC )
4		addsub4d.4	\|- ( ph -> D e. CC )
5	1 2	addcld	\|- ( ph -> ( A + B ) e. CC )
6	5 3 4	subsubd	\|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( ( A + B ) - C ) + D ) )
7	1 2 3	addsubd	\|- ( ph -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )
8	7	oveq1d	\|- ( ph -> ( ( ( A + B ) - C ) + D ) = ( ( ( A - C ) + B ) + D ) )
9	1 3	subcld	\|- ( ph -> ( A - C ) e. CC )
10	9 2 4	addassd	\|- ( ph -> ( ( ( A - C ) + B ) + D ) = ( ( A - C ) + ( B + D ) ) )
11	6 8 10	3eqtrd	\|- ( ph -> ( ( A + B ) - ( C - D ) ) = ( ( A - C ) + ( B + D ) ) )