Theorem csbopab

Description: Move substitution into a class abstraction. Version of csbopabw without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007) (Revised by NM, 23-Aug-2018)

	Ref	Expression
Assertion	csbopab	$$⊢⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		csbeq1	$$⊢w=A\to ⦋w/x⦌\left\{⟨y,z⟩|\phi \right\}=⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}$$
2		dfsbcq2	$$⊢w=A\to \left(\left[w/x\right]\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
3	2	opabbidv	$$⊢w=A\to \left\{⟨y,z⟩|\left[w/x\right]\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$
4	1 3	eqeq12d	$$⊢w=A\to \left(⦋w/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\left[w/x\right]\phi \right\}\leftrightarrow ⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}\right)$$
5		vex	$$⊢w\in \mathrm{V}$$
6		nfs1v	$$⊢Ⅎx\left[w/x\right]\phi$$
7	6	nfopab	$$⊢\underset{\_}{Ⅎ}x\left\{⟨y,z⟩|\left[w/x\right]\phi \right\}$$
8		sbequ12	$$⊢x=w\to \left(\phi \leftrightarrow \left[w/x\right]\phi \right)$$
9	8	opabbidv	$$⊢x=w\to \left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\left[w/x\right]\phi \right\}$$
10	5 7 9	csbief	$$⊢⦋w/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\left[w/x\right]\phi \right\}$$
11	4 10	vtoclg	$$⊢A\in \mathrm{V}\to ⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$
12		csbprc	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\varnothing$$
13		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$
14	13	con3i	$$⊢\neg A\in \mathrm{V}\to \neg \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
15	14	nexdv	$$⊢\neg A\in \mathrm{V}\to \neg \exists z\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
16	15	nexdv	$$⊢\neg A\in \mathrm{V}\to \neg \exists y\exists z\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
17		opabn0	$$⊢\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}\ne \varnothing \leftrightarrow \exists y\exists z\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
18	17	necon1bbii	$$⊢\neg \exists y\exists z\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}=\varnothing$$
19	16 18	sylib	$$⊢\neg A\in \mathrm{V}\to \left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}=\varnothing$$
20	12 19	eqtr4d	$$⊢\neg A\in \mathrm{V}\to ⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$
21	11 20	pm2.61i	$$⊢⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$