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Theorem sbcbr123 4446
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 3280 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 3376 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3376 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 3376 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 4409 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 3389 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 3389 . . . . 5  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 3389 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 4439 . . . 4  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 3382 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3382 . . . . 5  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 3382 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 4409 . . . 4  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 2173 . . 3  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 3118 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
16 sbcex 3287 . . . 4  |-  ( [. A  /  x ]. B R C  ->  A  e. 
_V )
1716con3i 135 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B R C )
18 br0 4441 . . . 4  |-  -.  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C
19 csbprc 3775 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ R  =  (/) )
2019breqd 4406 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C ) )
2118, 20mtbiri 301 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C )
2217, 212falsed 349 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
2315, 22pm2.61i 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 1405   [wsb 1763    e. wcel 1842   _Vcvv 3059   [.wsbc 3277   [_csb 3373   (/)c0 3738   class class class wbr 4395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396
This theorem is referenced by:  sbcbr  4447  sbcbr12g  4448  csbcnvgALT  5008  sbcfung  5592  csbfv12  5884  relowlpssretop  31281
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