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Theorem sbcbr123 4452
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Modified by NM, 22-Aug-2018.)
Assertion
Ref Expression
sbcbr123  |-  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C )

Proof of Theorem sbcbr123
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3297 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 3399 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3399 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 3399 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 4415 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 3412 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 3412 . . . . 5  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 3412 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 4445 . . . 4  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 3405 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3405 . . . . 5  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 3405 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 4415 . . . 4  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 2110 . . 3  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 3137 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
16 sbcex 3304 . . . 4  |-  ( [. A  /  x ]. B R C  ->  A  e. 
_V )
1716con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B R C )
18 noel 3750 . . . . 5  |-  -.  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >.  e.  (/)
19 df-br 4402 . . . . 5  |-  ( [_ A  /  x ]_ B (/) [_ A  /  x ]_ C  <->  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >.  e.  (/) )
2018, 19mtbir 299 . . . 4  |-  -.  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C
21 csbprc 3782 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ R  =  (/) )
2221breqd 4412 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C ) )
2320, 22mtbiri 303 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C )
2417, 232falsed 351 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
2515, 24pm2.61i 164 1  |-  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370   [wsb 1702    e. wcel 1758   _Vcvv 3078   [.wsbc 3294   [_csb 3396   (/)c0 3746   <.cop 3992   class class class wbr 4401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402
This theorem is referenced by:  sbcbr  4454  sbcbr12g  4455  csbcnvgALT  5133  sbcfung  5550  csbfv12  5835
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