Theorem joinlmuladdmuld

Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019)

	Ref	Expression
Hypotheses	joinlmuladdmuld.1	$$⊢\phi \to A\in ℂ$$
	joinlmuladdmuld.2	$$⊢\phi \to B\in ℂ$$
	joinlmuladdmuld.3	$$⊢\phi \to C\in ℂ$$
	joinlmuladdmuld.4	$$⊢\phi \to AB+CB=D$$
Assertion	joinlmuladdmuld	$$⊢\phi \to \left(A+C\right)B=D$$

Proof

Step	Hyp	Ref	Expression
1		joinlmuladdmuld.1	$$⊢\phi \to A\in ℂ$$
2		joinlmuladdmuld.2	$$⊢\phi \to B\in ℂ$$
3		joinlmuladdmuld.3	$$⊢\phi \to C\in ℂ$$
4		joinlmuladdmuld.4	$$⊢\phi \to AB+CB=D$$
5	1 3 2	adddird	$$⊢\phi \to \left(A+C\right)B=AB+CB$$
6	5 4	eqtrd	$$⊢\phi \to \left(A+C\right)B=D$$