Theorem add42d

Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addd.1	$$⊢\phi \to A\in ℂ$$
	addd.2	$$⊢\phi \to B\in ℂ$$
	addd.3	$$⊢\phi \to C\in ℂ$$
	add4d.4	$$⊢\phi \to D\in ℂ$$
Assertion	add42d	$$⊢\phi \to A+B+C+D=A+C+D+B$$

Proof

Step	Hyp	Ref	Expression
1		addd.1	$$⊢\phi \to A\in ℂ$$
2		addd.2	$$⊢\phi \to B\in ℂ$$
3		addd.3	$$⊢\phi \to C\in ℂ$$
4		add4d.4	$$⊢\phi \to D\in ℂ$$
5		add42	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(C\in ℂ\wedge D\in ℂ\right)\right)\to A+B+C+D=A+C+D+B$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to A+B+C+D=A+C+D+B$$