Theorem csbcog

Description: Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020)

	Ref	Expression
Assertion	csbcog	$$⊢A\in V\to ⦋A/x⦌\left(B\circ C\right)=⦋A/x⦌B\circ ⦋A/x⦌C$$

Proof

Step	Hyp	Ref	Expression
1		csbeq1	$$⊢y=A\to ⦋y/x⦌\left(B\circ C\right)=⦋A/x⦌\left(B\circ C\right)$$
2		csbeq1	$$⊢y=A\to ⦋y/x⦌B=⦋A/x⦌B$$
3		csbeq1	$$⊢y=A\to ⦋y/x⦌C=⦋A/x⦌C$$
4	2 3	coeq12d	$$⊢y=A\to ⦋y/x⦌B\circ ⦋y/x⦌C=⦋A/x⦌B\circ ⦋A/x⦌C$$
5	1 4	eqeq12d	$$⊢y=A\to \left(⦋y/x⦌\left(B\circ C\right)=⦋y/x⦌B\circ ⦋y/x⦌C\leftrightarrow ⦋A/x⦌\left(B\circ C\right)=⦋A/x⦌B\circ ⦋A/x⦌C\right)$$
6		vex	$$⊢y\in \mathrm{V}$$
7		nfcsb1v	$$⊢\underset{\_}{Ⅎ}x⦋y/x⦌B$$
8		nfcsb1v	$$⊢\underset{\_}{Ⅎ}x⦋y/x⦌C$$
9	7 8	nfco	$$⊢\underset{\_}{Ⅎ}x\left(⦋y/x⦌B\circ ⦋y/x⦌C\right)$$
10		csbeq1a	$$⊢x=y\to B=⦋y/x⦌B$$
11		csbeq1a	$$⊢x=y\to C=⦋y/x⦌C$$
12	10 11	coeq12d	$$⊢x=y\to B\circ C=⦋y/x⦌B\circ ⦋y/x⦌C$$
13	6 9 12	csbief	$$⊢⦋y/x⦌\left(B\circ C\right)=⦋y/x⦌B\circ ⦋y/x⦌C$$
14	5 13	vtoclg	$$⊢A\in V\to ⦋A/x⦌\left(B\circ C\right)=⦋A/x⦌B\circ ⦋A/x⦌C$$