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Metamath Proof Explorer

Theorem csbopabgALT

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) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion csbopabgALT 
|- ( A e. V -> [_ 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 } )