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Theorem csbov 6314
Description: Move class substitution in and out of an operation. (Contributed by NM, 23-Aug-2018.)
Assertion
Ref Expression
csbov  |-  [_ A  /  x ]_ ( B F C )  =  ( B [_ A  /  x ]_ F C )
Distinct variable groups:    x, B    x, C
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem csbov
StepHypRef Expression
1 csbov123 6313 . 2  |-  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
)
2 csbconstg 3448 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
3 csbconstg 3448 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ C  =  C )
42, 3oveq12d 6300 . . 3  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( B [_ A  /  x ]_ F C ) )
5 0fv 5897 . . . . 5  |-  ( (/) ` 
<. B ,  C >. )  =  (/)
6 df-ov 6285 . . . . 5  |-  ( B
(/) C )  =  ( (/) `  <. B ,  C >. )
7 df-ov 6285 . . . . . 6  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  (
(/) `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )
8 0fv 5897 . . . . . 6  |-  ( (/) ` 
<. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )  =  (/)
97, 8eqtri 2496 . . . . 5  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  (/)
105, 6, 93eqtr4ri 2507 . . . 4  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  ( B (/) C )
11 csbprc 3821 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ F  =  (/) )
1211oveqd 6299 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C ) )
1311oveqd 6299 . . . 4  |-  ( -.  A  e.  _V  ->  ( B [_ A  /  x ]_ F C )  =  ( B (/) C ) )
1410, 12, 133eqtr4a 2534 . . 3  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( B [_ A  /  x ]_ F C ) )
154, 14pm2.61i 164 . 2  |-  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( B [_ A  /  x ]_ F C )
161, 15eqtri 2496 1  |-  [_ A  /  x ]_ ( B F C )  =  ( B [_ A  /  x ]_ F C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435   (/)c0 3785   <.cop 4033   ` cfv 5586  (class class class)co 6282
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 4576  ax-pow 4625
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009  df-iota 5549  df-fv 5594  df-ov 6285
This theorem is referenced by:  mptcoe1matfsupp  19070  mp2pm2mplem4  19077
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