Theorem csbin

Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	csbin	$$⊢⦋A/x⦌\left(B\cap C\right)=⦋A/x⦌B\cap ⦋A/x⦌C$$

Proof

Step	Hyp	Ref	Expression
1		csbeq1	$$⊢y=A\to ⦋y/x⦌\left(B\cap C\right)=⦋A/x⦌\left(B\cap C\right)$$
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	ineq12d	$$⊢y=A\to ⦋y/x⦌B\cap ⦋y/x⦌C=⦋A/x⦌B\cap ⦋A/x⦌C$$
5	1 4	eqeq12d	$$⊢y=A\to \left(⦋y/x⦌\left(B\cap C\right)=⦋y/x⦌B\cap ⦋y/x⦌C\leftrightarrow ⦋A/x⦌\left(B\cap C\right)=⦋A/x⦌B\cap ⦋A/x⦌C\right)$$
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	nfin	$$⊢\underset{\_}{Ⅎ}x\left(⦋y/x⦌B\cap ⦋y/x⦌C\right)$$
10		csbeq1a	$$⊢x=y\to B=⦋y/x⦌B$$
11		csbeq1a	$$⊢x=y\to C=⦋y/x⦌C$$
12	10 11	ineq12d	$$⊢x=y\to B\cap C=⦋y/x⦌B\cap ⦋y/x⦌C$$
13	6 9 12	csbief	$$⊢⦋y/x⦌\left(B\cap C\right)=⦋y/x⦌B\cap ⦋y/x⦌C$$
14	5 13	vtoclg	$$⊢A\in \mathrm{V}\to ⦋A/x⦌\left(B\cap C\right)=⦋A/x⦌B\cap ⦋A/x⦌C$$
15		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left(B\cap C\right)=\varnothing$$
16		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌B=\varnothing$$
17		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌C=\varnothing$$
18	16 17	ineq12d	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌B\cap ⦋A/x⦌C=\varnothing \cap \varnothing$$
19		in0	$$⊢\varnothing \cap \varnothing =\varnothing$$
20	18 19	eqtr2di	$$⊢\neg A\in \mathrm{V}\to \varnothing =⦋A/x⦌B\cap ⦋A/x⦌C$$
21	15 20	eqtrd	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left(B\cap C\right)=⦋A/x⦌B\cap ⦋A/x⦌C$$
22	14 21	pm2.61i	$$⊢⦋A/x⦌\left(B\cap C\right)=⦋A/x⦌B\cap ⦋A/x⦌C$$