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Metamath Proof Explorer

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 }