Theorem joinlmulsubmuld

Description: Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018)

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

Proof

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