Theorem addsubeq4d

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

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
	subaddd.3	$$⊢\phi \to C\in ℂ$$
	addsub4d.4	$$⊢\phi \to D\in ℂ$$
Assertion	addsubeq4d	$$⊢\phi \to \left(A+B=C+D\leftrightarrow C-A=B-D\right)$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		subaddd.3	$$⊢\phi \to C\in ℂ$$
4		addsub4d.4	$$⊢\phi \to D\in ℂ$$
5		addsubeq4	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(C\in ℂ\wedge D\in ℂ\right)\right)\to \left(A+B=C+D\leftrightarrow C-A=B-D\right)$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to \left(A+B=C+D\leftrightarrow C-A=B-D\right)$$