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Theorem sbcbr123 4498
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 3334 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 3438 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3438 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 3438 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 4461 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 3451 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 3451 . . . . 5  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 3451 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 4491 . . . 4  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 3444 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3444 . . . . 5  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 3444 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 4461 . . . 4  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 2123 . . 3  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 3172 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
16 sbcex 3341 . . . 4  |-  ( [. A  /  x ]. B R C  ->  A  e. 
_V )
1716con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B R C )
18 noel 3789 . . . . 5  |-  -.  <. [_ A  /  x ]_ B ,  [_ A  /  x ]_ C >.  e.  (/)
19 df-br 4448 . . . . 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 3821 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ R  =  (/) )
2221breqd 4458 . . . 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 1379   [wsb 1711    e. wcel 1767   _Vcvv 3113   [.wsbc 3331   [_csb 3435   (/)c0 3785   <.cop 4033   class class class wbr 4447
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  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-br 4448
This theorem is referenced by:  sbcbr  4500  sbcbr12g  4501  csbcnvgALT  5185  sbcfung  5609  csbfv12  5899
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