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Theorem sbcbr123 4470
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 3282 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] B R C  <->  [. A  /  x ]. B R C ) )
2 csbeq1 3378 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 csbeq1 3378 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ R  = 
[_ A  /  x ]_ R )
4 csbeq1 3378 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4breq123d 4432 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
6 nfcsb1v 3391 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
7 nfcsb1v 3391 . . . . 5  |-  F/_ x [_ y  /  x ]_ R
8 nfcsb1v 3391 . . . . 5  |-  F/_ x [_ y  /  x ]_ C
96, 7, 8nfbr 4463 . . . 4  |-  F/ x [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
10 csbeq1a 3384 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
11 csbeq1a 3384 . . . . 5  |-  ( x  =  y  ->  R  =  [_ y  /  x ]_ R )
12 csbeq1a 3384 . . . . 5  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1310, 11, 12breq123d 4432 . . . 4  |-  ( x  =  y  ->  ( B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
) )
149, 13sbie 2248 . . 3  |-  ( [ y  /  x ] B R C  <->  [_ y  /  x ]_ B [_ y  /  x ]_ R [_ y  /  x ]_ C
)
151, 5, 14vtoclbg 3120 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
16 sbcex 3289 . . . 4  |-  ( [. A  /  x ]. B R C  ->  A  e. 
_V )
1716con3i 142 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. B R C )
18 br0 4465 . . . 4  |-  -.  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C
19 csbprc 3782 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ R  =  (/) )
2019breqd 4429 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B (/) [_ A  /  x ]_ C ) )
2118, 20mtbiri 309 . . 3  |-  ( -.  A  e.  _V  ->  -. 
[_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C )
2217, 212falsed 357 . 2  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. B R C  <->  [_ A  /  x ]_ B [_ A  /  x ]_ R [_ A  /  x ]_ C
) )
2315, 22pm2.61i 169 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 189    = wceq 1455   [wsb 1808    e. wcel 1898   _Vcvv 3057   [.wsbc 3279   [_csb 3375   (/)c0 3743   class class class wbr 4418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419
This theorem is referenced by:  sbcbr  4471  sbcbr12g  4472  csbcnvgALT  5041  sbcfung  5628  csbfv12  5927  relowlpssretop  31813
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