Theorem int-sqdefd

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

	Ref	Expression
Hypotheses	int-sqdefd.1	$$⊢\phi \to B\in ℝ$$
	int-sqdefd.2	$$⊢\phi \to A=B$$
Assertion	int-sqdefd	$$⊢\phi \to AB=A^2$$

Proof

Step	Hyp	Ref	Expression
1		int-sqdefd.1	$$⊢\phi \to B\in ℝ$$
2		int-sqdefd.2	$$⊢\phi \to A=B$$
3	2	oveq1d	$$⊢\phi \to A^2=B^2$$
4	1	recnd	$$⊢\phi \to B\in ℂ$$
5	4	sqvald	$$⊢\phi \to B^2=BB$$
6		eqcom	$$⊢A=B\leftrightarrow B=A$$
7	6	imbi2i	$$⊢\left(\phi \to A=B\right)\leftrightarrow \left(\phi \to B=A\right)$$
8	2 7	mpbi	$$⊢\phi \to B=A$$
9	8	oveq1d	$$⊢\phi \to BB=AB$$
10	5 9	eqtrd	$$⊢\phi \to B^2=AB$$
11	3 10	eqtrd	$$⊢\phi \to A^2=AB$$
12	11	eqcomd	$$⊢\phi \to AB=A^2$$