Metamath Proof Explorer

Theorem csbriota

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013) (Revised by NM, 2-Sep-2018)

Ref Expression
Assertion csbriota 
|- [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph )

Proof

Step Hyp Ref Expression
1 csbeq1 
 |-  ( z = A -> [_ z / x ]_ ( iota_ y e. B ph ) = [_ A / x ]_ ( iota_ y e. B ph ) )
2 dfsbcq2 
 |-  ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3 2 riotabidv 
 |-  ( z = A -> ( iota_ y e. B [ z / x ] ph ) = ( iota_ y e. B [. A / x ]. ph ) )
4 1 3 eqeq12d 
 |-  ( z = A -> ( [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) <-> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) ) )
5 vex 
 |-  z e. _V
6 nfs1v 
 |-  F/ x [ z / x ] ph
7 nfcv 
 |-  F/_ x B
8 6 7 nfriota 
 |-  F/_ x ( iota_ y e. B [ z / x ] ph )
9 sbequ12 
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
10 9 riotabidv 
 |-  ( x = z -> ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph ) )
11 5 8 10 csbief 
 |-  [_ z / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [ z / x ] ph )
12 4 11 vtoclg 
 |-  ( A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
13 csbprc 
 |-  ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = (/) )
14 df-riota 
 |-  ( iota_ y e. B [. A / x ]. ph ) = ( iota y ( y e. B /\ [. A / x ]. ph ) )
15 euex 
 |-  ( E! y ( y e. B /\ [. A / x ]. ph ) -> E. y ( y e. B /\ [. A / x ]. ph ) )
16 sbcex 
 |-  ( [. A / x ]. ph -> A e. _V )
17 16 adantl 
 |-  ( ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
18 17 exlimiv 
 |-  ( E. y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
19 15 18 syl 
 |-  ( E! y ( y e. B /\ [. A / x ]. ph ) -> A e. _V )
20 iotanul 
 |-  ( -. E! y ( y e. B /\ [. A / x ]. ph ) -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
21 19 20 nsyl5 
 |-  ( -. A e. _V -> ( iota y ( y e. B /\ [. A / x ]. ph ) ) = (/) )
22 14 21 eqtr2id 
 |-  ( -. A e. _V -> (/) = ( iota_ y e. B [. A / x ]. ph ) )
23 13 22 eqtrd 
 |-  ( -. A e. _V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
24 12 23 pm2.61i 
 |-  [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph )