Theorem csbopab

Description: Move substitution into a class abstraction. Version of csbopabgALT 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 ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }

Proof

Step	Hyp	Ref	Expression
1		csbeq1	|- ( w = A -> [_ w / x ]_ { <. y , z >. | ph } = [_ A / x ]_ { <. y , z >. | ph } )
2		dfsbcq2	|- ( w = A -> ( [ w / x ] ph <-> [. A / x ]. ph ) )
3	2	opabbidv	|- ( w = A -> { <. y , z >. | [ w / x ] ph } = { <. y , z >. | [. A / x ]. ph } )
4	1 3	eqeq12d	|- ( w = A -> ( [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } <-> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } ) )
5		vex	|- w e. _V
6		nfs1v	|- F/ x [ w / x ] ph
7	6	nfopab	|- F/_ x { <. y , z >. | [ w / x ] ph }
8		sbequ12	|- ( x = w -> ( ph <-> [ w / x ] ph ) )
9	8	opabbidv	|- ( x = w -> { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph } )
10	5 7 9	csbief	|- [_ w / x ]_ { <. y , z >. | ph } = { <. y , z >. | [ w / x ] ph }
11	4 10	vtoclg	|- ( A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
12		csbprc	|- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = (/) )
13		sbcex	|- ( [. A / x ]. ph -> A e. _V )
14	13	con3i	|- ( -. A e. _V -> -. [. A / x ]. ph )
15	14	nexdv	|- ( -. A e. _V -> -. E. z [. A / x ]. ph )
16	15	nexdv	|- ( -. A e. _V -> -. E. y E. z [. A / x ]. ph )
17		opabn0	|- ( { <. y , z >. | [. A / x ]. ph } =/= (/) <-> E. y E. z [. A / x ]. ph )
18	17	necon1bbii	|- ( -. E. y E. z [. A / x ]. ph <-> { <. y , z >. | [. A / x ]. ph } = (/) )
19	16 18	sylib	|- ( -. A e. _V -> { <. y , z >. | [. A / x ]. ph } = (/) )
20	12 19	eqtr4d	|- ( -. A e. _V -> [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph } )
21	11 20	pm2.61i	|- [_ A / x ]_ { <. y , z >. | ph } = { <. y , z >. | [. A / x ]. ph }