Theorem csbopabw

Description: Move substitution into a class abstraction. Version of csbopab with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007) (Proof shortened by Mario Carneiro, 17-Nov-2016)

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

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	$$⊢A\in V\to ⦋A/x⦌\left\{⟨y,z⟩|\phi \right\}=\left\{⟨y,z⟩|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right\}$$