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Metamath Proof Explorer

Theorem csbiota

Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017) (Revised by NM, 23-Aug-2018)

Ref Expression
Assertion csbiota 
|- [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph )

Proof

Step Hyp Ref Expression
1 csbeq1 
 |-  ( z = A -> [_ z / x ]_ ( iota y ph ) = [_ A / x ]_ ( iota y ph ) )
2 dfsbcq2 
 |-  ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3 2 iotabidv 
 |-  ( z = A -> ( iota y [ z / x ] ph ) = ( iota y [. A / x ]. ph ) )
4 1 3 eqeq12d 
 |-  ( z = A -> ( [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph ) <-> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) ) )
5 vex 
 |-  z e. _V
6 nfs1v 
 |-  F/ x [ z / x ] ph
7 6 nfiotaw 
 |-  F/_ x ( iota y [ z / x ] ph )
8 sbequ12 
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
9 8 iotabidv 
 |-  ( x = z -> ( iota y ph ) = ( iota y [ z / x ] ph ) )
10 5 7 9 csbief 
 |-  [_ z / x ]_ ( iota y ph ) = ( iota y [ z / x ] ph )
11 4 10 vtoclg 
 |-  ( A e. _V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )
12 csbprc 
 |-  ( -. A e. _V -> [_ A / x ]_ ( iota y 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. y [. A / x ]. ph )
16 euex 
 |-  ( E! y [. A / x ]. ph -> E. y [. A / x ]. ph )
17 16 con3i 
 |-  ( -. E. y [. A / x ]. ph -> -. E! y [. A / x ]. ph )
18 iotanul 
 |-  ( -. E! y [. A / x ]. ph -> ( iota y [. A / x ]. ph ) = (/) )
19 15 17 18 3syl 
 |-  ( -. A e. _V -> ( iota y [. A / x ]. ph ) = (/) )
20 12 19 eqtr4d 
 |-  ( -. A e. _V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )
21 11 20 pm2.61i 
 |-  [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph )