Theorem mul4d

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

	Ref	Expression
Hypotheses	muld.1	|- ( ph -> A e. CC )
	addcomd.2	|- ( ph -> B e. CC )
	addcand.3	|- ( ph -> C e. CC )
	mul4d.4	|- ( ph -> D e. CC )
Assertion	mul4d	|- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )

Proof

Step	Hyp	Ref	Expression
1		muld.1	|- ( ph -> A e. CC )
2		addcomd.2	|- ( ph -> B e. CC )
3		addcand.3	|- ( ph -> C e. CC )
4		mul4d.4	|- ( ph -> D e. CC )
5		mul4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )