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Theorem csbov 6231
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 6230 . 2  |-  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
)
2 csbconstg 3361 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ B  =  B )
3 csbconstg 3361 . . . 4  |-  ( A  e.  _V  ->  [_ A  /  x ]_ C  =  C )
42, 3oveq12d 6214 . . 3  |-  ( A  e.  _V  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( B [_ A  /  x ]_ F C ) )
5 0fv 5807 . . . . 5  |-  ( (/) ` 
<. B ,  C >. )  =  (/)
6 df-ov 6199 . . . . 5  |-  ( B
(/) C )  =  ( (/) `  <. B ,  C >. )
7 df-ov 6199 . . . . . 6  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  (
(/) `  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )
8 0fv 5807 . . . . . 6  |-  ( (/) ` 
<. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >. )  =  (/)
97, 8eqtri 2411 . . . . 5  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  (/)
105, 6, 93eqtr4ri 2422 . . . 4  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C )  =  ( B (/) C )
11 csbprc 3748 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ F  =  (/) )
1211oveqd 6213 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C )  =  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C ) )
1311oveqd 6213 . . . 4  |-  ( -.  A  e.  _V  ->  ( B [_ A  /  x ]_ F C )  =  ( B (/) C ) )
1410, 12, 133eqtr4a 2449 . . 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 2411 1  |-  [_ A  /  x ]_ ( B F C )  =  ( B [_ A  /  x ]_ F C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1399    e. wcel 1826   _Vcvv 3034   [_csb 3348   (/)c0 3711   <.cop 3950   ` cfv 5496  (class class class)co 6196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-nul 4496  ax-pow 4543
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-dm 4923  df-iota 5460  df-fv 5504  df-ov 6199
This theorem is referenced by:  mptcoe1matfsupp  19388  mp2pm2mplem4  19395
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