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Theorem csbin 3863
Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbin  |-  [_ A  /  x ]_ ( B  i^i  C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C )

Proof of Theorem csbin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3433 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B  i^i  C )  = 
[_ A  /  x ]_ ( B  i^i  C
) )
2 csbeq1 3433 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3433 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
42, 3ineq12d 3697 . . . 4  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B  i^i  [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C ) )
51, 4eqeq12d 2479 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( B  i^i  C
)  =  ( [_ y  /  x ]_ B  i^i  [_ y  /  x ]_ C )  <->  [_ A  /  x ]_ ( B  i^i  C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C ) ) )
6 vex 3112 . . . 4  |-  y  e. 
_V
7 nfcsb1v 3446 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
8 nfcsb1v 3446 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
97, 8nfin 3701 . . . 4  |-  F/_ x
( [_ y  /  x ]_ B  i^i  [_ y  /  x ]_ C )
10 csbeq1a 3439 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3439 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1210, 11ineq12d 3697 . . . 4  |-  ( x  =  y  ->  ( B  i^i  C )  =  ( [_ y  /  x ]_ B  i^i  [_ y  /  x ]_ C ) )
136, 9, 12csbief 3455 . . 3  |-  [_ y  /  x ]_ ( B  i^i  C )  =  ( [_ y  /  x ]_ B  i^i  [_ y  /  x ]_ C )
145, 13vtoclg 3167 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( B  i^i  C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C ) )
15 csbprc 3830 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( B  i^i  C )  =  (/) )
16 csbprc 3830 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
17 csbprc 3830 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ C  =  (/) )
1816, 17ineq12d 3697 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C )  =  ( (/)  i^i  (/) ) )
19 in0 3820 . . . 4  |-  ( (/)  i^i  (/) )  =  (/)
2018, 19syl6req 2515 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C ) )
2115, 20eqtrd 2498 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( B  i^i  C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C ) )
2214, 21pm2.61i 164 1  |-  [_ A  /  x ]_ ( B  i^i  C )  =  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   [_csb 3430    i^i cin 3470   (/)c0 3793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794
This theorem is referenced by:  csbres  5286  disjxpin  27587  onfrALTlem5  33457  onfrALTlem4  33458
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