Theorem mul4d

Description: Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	muld.1	$$⊢\phi \to A\in ℂ$$
	addcomd.2	$$⊢\phi \to B\in ℂ$$
	addcand.3	$$⊢\phi \to C\in ℂ$$
	mul4d.4	$$⊢\phi \to D\in ℂ$$
Assertion	mul4d	$$⊢\phi \to ABCD=ACBD$$

Proof

Step	Hyp	Ref	Expression
1		muld.1	$$⊢\phi \to A\in ℂ$$
2		addcomd.2	$$⊢\phi \to B\in ℂ$$
3		addcand.3	$$⊢\phi \to C\in ℂ$$
4		mul4d.4	$$⊢\phi \to D\in ℂ$$
5		mul4	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(C\in ℂ\wedge D\in ℂ\right)\right)\to ABCD=ACBD$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to ABCD=ACBD$$