Theorem isoeq145d

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

	Ref	Expression
Hypotheses	isoeq145.1	|- ( ph -> F = G )
	isoeq145.4	|- ( ph -> A = C )
	isoeq145.5	|- ( ph -> B = D )
Assertion	isoeq145d	|- ( ph -> ( F Isom R , S ( A , B ) <-> G Isom R , S ( C , D ) ) )

Proof

Step	Hyp	Ref	Expression
1		isoeq145.1	|- ( ph -> F = G )
2		isoeq145.4	|- ( ph -> A = C )
3		isoeq145.5	|- ( ph -> B = D )
4		isoeq1	|- ( F = G -> ( F Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )
5	1 4	syl	|- ( ph -> ( F Isom R , S ( A , B ) <-> G Isom R , S ( A , B ) ) )
6		isoeq4	|- ( A = C -> ( G Isom R , S ( A , B ) <-> G Isom R , S ( C , B ) ) )
7	2 6	syl	|- ( ph -> ( G Isom R , S ( A , B ) <-> G Isom R , S ( C , B ) ) )
8		isoeq5	|- ( B = D -> ( G Isom R , S ( C , B ) <-> G Isom R , S ( C , D ) ) )
9	3 8	syl	|- ( ph -> ( G Isom R , S ( C , B ) <-> G Isom R , S ( C , D ) ) )
10	5 7 9	3bitrd	|- ( ph -> ( F Isom R , S ( A , B ) <-> G Isom R , S ( C , D ) ) )