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Theorem sbcbr123 4671
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]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)

Proof of Theorem sbcbr123
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3432 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 ∈ V)
2 br0 4666 . . . 4 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
3 csbprc 3957 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
43breqd 4629 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
52, 4mtbiri 317 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
65con4i 113 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 ∈ V)
7 dfsbcq2 3425 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
8 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
9 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
10 csbeq1 3522 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
118, 9, 10breq123d 4632 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
12 nfcsb1v 3535 . . . . 5 𝑥𝑦 / 𝑥𝐵
13 nfcsb1v 3535 . . . . 5 𝑥𝑦 / 𝑥𝑅
14 nfcsb1v 3535 . . . . 5 𝑥𝑦 / 𝑥𝐶
1512, 13, 14nfbr 4664 . . . 4 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
16 csbeq1a 3528 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
17 csbeq1a 3528 . . . . 5 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
18 csbeq1a 3528 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1916, 17, 18breq123d 4632 . . . 4 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
2015, 19sbie 2412 . . 3 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
217, 11, 20vtoclbg 3258 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
221, 6, 21pm5.21nii 368 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  [wsb 1882  wcel 1992  Vcvv 3191  [wsbc 3422  csb 3519  c0 3896   class class class wbr 4618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619
This theorem is referenced by:  sbcbr  4672  sbcbr12g  4673  csbcnvgALT  5272  sbcfung  5873  csbfv12  6189  relowlpssretop  32836
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