Theorem csbun

Description: Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015) (Revised by Mario Carneiro, 11-Dec-2016) (Revised by NM, 13-Sep-2018)

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

Proof

Step	Hyp	Ref	Expression
1		csbeq1	$$⊢y=A\to ⦋y/x⦌\left(B\cup C\right)=⦋A/x⦌\left(B\cup C\right)$$
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	uneq12d	$$⊢y=A\to ⦋y/x⦌B\cup ⦋y/x⦌C=⦋A/x⦌B\cup ⦋A/x⦌C$$
5	1 4	eqeq12d	$$⊢y=A\to \left(⦋y/x⦌\left(B\cup C\right)=⦋y/x⦌B\cup ⦋y/x⦌C\leftrightarrow ⦋A/x⦌\left(B\cup C\right)=⦋A/x⦌B\cup ⦋A/x⦌C\right)$$
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	nfun	$$⊢\underset{\_}{Ⅎ}x\left(⦋y/x⦌B\cup ⦋y/x⦌C\right)$$
10		csbeq1a	$$⊢x=y\to B=⦋y/x⦌B$$
11		csbeq1a	$$⊢x=y\to C=⦋y/x⦌C$$
12	10 11	uneq12d	$$⊢x=y\to B\cup C=⦋y/x⦌B\cup ⦋y/x⦌C$$
13	6 9 12	csbief	$$⊢⦋y/x⦌\left(B\cup C\right)=⦋y/x⦌B\cup ⦋y/x⦌C$$
14	5 13	vtoclg	$$⊢A\in \mathrm{V}\to ⦋A/x⦌\left(B\cup C\right)=⦋A/x⦌B\cup ⦋A/x⦌C$$
15		un0	$$⊢\varnothing \cup \varnothing =\varnothing$$
16	15	a1i	$$⊢\neg A\in \mathrm{V}\to \varnothing \cup \varnothing =\varnothing$$
17		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌B=\varnothing$$
18		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌C=\varnothing$$
19	17 18	uneq12d	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌B\cup ⦋A/x⦌C=\varnothing \cup \varnothing$$
20		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left(B\cup C\right)=\varnothing$$
21	16 19 20	3eqtr4rd	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left(B\cup C\right)=⦋A/x⦌B\cup ⦋A/x⦌C$$
22	14 21	pm2.61i	$$⊢⦋A/x⦌\left(B\cup C\right)=⦋A/x⦌B\cup ⦋A/x⦌C$$