Theorem addsubeq4d

Description: Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016)

	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	addsubeq4d	|- ( ph -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( 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		addsubeq4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )