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Theorem csbfv12 5907
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv12  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )

Proof of Theorem csbfv12
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbiota 5586 . . . 4  |-  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [. A  /  x ]. B F y )
2 sbcbr123 4504 . . . . . 6  |-  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y
)
3 csbconstg 3453 . . . . . . 7  |-  ( A  e.  _V  ->  [_ A  /  x ]_ y  =  y )
43breq2d 4465 . . . . . 6  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
52, 4syl5bb 257 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
65iotabidv 5578 . . . 4  |-  ( A  e.  _V  ->  ( iota y [. A  /  x ]. B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
71, 6syl5eq 2520 . . 3  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
8 df-fv 5602 . . . 4  |-  ( F `
 B )  =  ( iota y B F y )
98csbeq2i 3841 . . 3  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ ( iota y B F y )
10 df-fv 5602 . . 3  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F
y )
117, 9, 103eqtr4g 2533 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
12 csbprc 3826 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  B )  =  (/) )
13 csbprc 3826 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ F  =  (/) )
1413fveq1d 5874 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( (/) `  [_ A  /  x ]_ B ) )
15 0fv 5905 . . . 4  |-  ( (/) ` 
[_ A  /  x ]_ B )  =  (/)
1614, 15syl6req 2525 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
1712, 16eqtrd 2508 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
1811, 17pm2.61i 164 1  |-  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   [.wsbc 3336   [_csb 3440   (/)c0 3790   class class class wbr 4453   iotacio 5555   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582  ax-pow 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602
This theorem is referenced by:  csbfv2g  5909  coe1fzgsumdlem  18213  evl1gsumdlem  18262  cdlemk42  36138
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