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Theorem sbcbr123 5111
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
sbcbr123 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)

Proof of Theorem sbcbr123
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 ∈ V)
2 br0 5106 . . . 4 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
3 csbprc 4355 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
43breqd 5068 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
52, 4mtbiri 328 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
65con4i 114 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 ∈ V)
7 dfsbcq2 3772 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
8 csbeq1 3883 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
9 csbeq1 3883 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
10 csbeq1 3883 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
118, 9, 10breq123d 5071 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
12 nfcsb1v 3904 . . . . 5 𝑥𝑦 / 𝑥𝐵
13 nfcsb1v 3904 . . . . 5 𝑥𝑦 / 𝑥𝑅
14 nfcsb1v 3904 . . . . 5 𝑥𝑦 / 𝑥𝐶
1512, 13, 14nfbr 5104 . . . 4 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
16 csbeq1a 3894 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
17 csbeq1a 3894 . . . . 5 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
18 csbeq1a 3894 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1916, 17, 18breq123d 5071 . . . 4 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
2015, 19sbiev 2321 . . 3 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
217, 11, 20vtoclbg 3566 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
221, 6, 21pm5.21nii 380 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  [wsb 2060  wcel 2105  Vcvv 3492  [wsbc 3769  csb 3880  c0 4288   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058
This theorem is referenced by:  sbcbr  5112  sbcbr12g  5113  csbcnvgALT  5748  sbcfung  6372  csbfv12  6706  relowlpssretop  34527
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