Theorem int-mulassocd

Description: MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-mulassocd.1	$$⊢\phi \to B\in ℝ$$
	int-mulassocd.2	$$⊢\phi \to C\in ℝ$$
	int-mulassocd.3	$$⊢\phi \to D\in ℝ$$
	int-mulassocd.4	$$⊢\phi \to A=B$$
Assertion	int-mulassocd	$$⊢\phi \to BCD=ACD$$

Proof

Step	Hyp	Ref	Expression
1		int-mulassocd.1	$$⊢\phi \to B\in ℝ$$
2		int-mulassocd.2	$$⊢\phi \to C\in ℝ$$
3		int-mulassocd.3	$$⊢\phi \to D\in ℝ$$
4		int-mulassocd.4	$$⊢\phi \to A=B$$
5	1	recnd	$$⊢\phi \to B\in ℂ$$
6	2	recnd	$$⊢\phi \to C\in ℂ$$
7	3	recnd	$$⊢\phi \to D\in ℂ$$
8	5 6 7	mulassd	$$⊢\phi \to BCD=BCD$$
9	4	eqcomd	$$⊢\phi \to B=A$$
10	9	oveq1d	$$⊢\phi \to BC=AC$$
11	10	oveq1d	$$⊢\phi \to BCD=ACD$$
12	8 11	eqtr3d	$$⊢\phi \to BCD=ACD$$