MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbov123 Structured version   Visualization version   Unicode version

Theorem csbov123 6342
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Revised by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbov123  |-  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
)

Proof of Theorem csbov123
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B F C )  = 
[_ A  /  x ]_ ( B F C ) )
2 csbeq1 3352 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3352 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3352 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4oveq123d 6329 . . . 4  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
61, 5eqeq12d 2486 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  <->  [_ A  /  x ]_ ( B F C )  =  (
[_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) ) )
7 vex 3034 . . . 4  |-  y  e. 
_V
8 nfcsb1v 3365 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3365 . . . . 5  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3365 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfov 6334 . . . 4  |-  F/_ x
( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )
12 csbeq1a 3358 . . . . 5  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3358 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3358 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14oveq123d 6329 . . . 4  |-  ( x  =  y  ->  ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
) )
167, 11, 15csbief 3374 . . 3  |-  [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
)
176, 16vtoclg 3093 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
) )
18 csbprc 3774 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( B F C )  =  (/) )
19 df-ov 6311 . . . 4  |-  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( [_ A  /  x ]_ F `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )
20 csbprc 3774 . . . . . 6  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ F  =  (/) )
2120fveq1d 5881 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )  =  (
(/) `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. ) )
22 0fv 5912 . . . . 5  |-  ( (/) ` 
<. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )  =  (/)
2321, 22syl6eq 2521 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )  =  (/) )
2419, 23syl5req 2518 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
2518, 24eqtrd 2505 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
2617, 25pm2.61i 169 1  |-  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   [_csb 3349   (/)c0 3722   <.cop 3965   ` cfv 5589  (class class class)co 6308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527  ax-pow 4579
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-dm 4849  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by:  csbov  6343  csbov12g  6344  relowlpssretop  31837  rdgeqoa  31843
  Copyright terms: Public domain W3C validator