Theorem resisoeq45d

Description: Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025)

	Ref	Expression
Hypotheses	resisoeq45.4	|- ( ph -> A = C )
	resisoeq45.5	|- ( ph -> B = D )
Assertion	resisoeq45d	|- ( ph -> ( ( F |` A ) Isom R , S ( A , B ) <-> ( F |` C ) Isom R , S ( C , D ) ) )

Proof

Step	Hyp	Ref	Expression
1		resisoeq45.4	|- ( ph -> A = C )
2		resisoeq45.5	|- ( ph -> B = D )
3	1	reseq2d	|- ( ph -> ( F |` A ) = ( F |` C ) )
4	3 1 2	isoeq145d	|- ( ph -> ( ( F |` A ) Isom R , S ( A , B ) <-> ( F |` C ) Isom R , S ( C , D ) ) )