Theorem isoeq145d

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

	Ref	Expression
Hypotheses	isoeq145.1	$$⊢\phi \to F=G$$
	isoeq145.4	$$⊢\phi \to A=C$$
	isoeq145.5	$$⊢\phi \to B=D$$
Assertion	isoeq145d	$$⊢\phi \to \left(F{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,D\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		isoeq145.1	$$⊢\phi \to F=G$$
2		isoeq145.4	$$⊢\phi \to A=C$$
3		isoeq145.5	$$⊢\phi \to B=D$$
4		isoeq1	$$⊢F=G\to \left(F{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(A,B\right)\right)$$
5	1 4	syl	$$⊢\phi \to \left(F{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(A,B\right)\right)$$
6		isoeq4	$$⊢A=C\to \left(G{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,B\right)\right)$$
7	2 6	syl	$$⊢\phi \to \left(G{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,B\right)\right)$$
8		isoeq5	$$⊢B=D\to \left(G{Isom}_{R,S}\left(C,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,D\right)\right)$$
9	3 8	syl	$$⊢\phi \to \left(G{Isom}_{R,S}\left(C,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,D\right)\right)$$
10	5 7 9	3bitrd	$$⊢\phi \to \left(F{Isom}_{R,S}\left(A,B\right)\leftrightarrow G{Isom}_{R,S}\left(C,D\right)\right)$$