Theorem int-sqdefd

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

	Ref	Expression
Hypotheses	int-sqdefd.1	|- ( ph -> B e. RR )
	int-sqdefd.2	|- ( ph -> A = B )
Assertion	int-sqdefd	|- ( ph -> ( A x. B ) = ( A ^ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		int-sqdefd.1	|- ( ph -> B e. RR )
2		int-sqdefd.2	|- ( ph -> A = B )
3	2	oveq1d	|- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
4	1	recnd	|- ( ph -> B e. CC )
5	4	sqvald	|- ( ph -> ( B ^ 2 ) = ( B x. B ) )
6		eqcom	|- ( A = B <-> B = A )
7	6	imbi2i	|- ( ( ph -> A = B ) <-> ( ph -> B = A ) )
8	2 7	mpbi	|- ( ph -> B = A )
9	8	oveq1d	|- ( ph -> ( B x. B ) = ( A x. B ) )
10	5 9	eqtrd	|- ( ph -> ( B ^ 2 ) = ( A x. B ) )
11	3 10	eqtrd	|- ( ph -> ( A ^ 2 ) = ( A x. B ) )
12	11	eqcomd	|- ( ph -> ( A x. B ) = ( A ^ 2 ) )