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Theorem sbcbr123 5016
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 3716 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 ∈ V)
2 br0 5011 . . . 4 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
3 csbprc 4278 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
43breqd 4973 . . . 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 3709 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
8 csbeq1 3814 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
9 csbeq1 3814 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
10 csbeq1 3814 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
118, 9, 10breq123d 4976 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
12 nfcsb1v 3833 . . . . 5 𝑥𝑦 / 𝑥𝐵
13 nfcsb1v 3833 . . . . 5 𝑥𝑦 / 𝑥𝑅
14 nfcsb1v 3833 . . . . 5 𝑥𝑦 / 𝑥𝐶
1512, 13, 14nfbr 5009 . . . 4 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
16 csbeq1a 3824 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
17 csbeq1a 3824 . . . . 5 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
18 csbeq1a 3824 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1916, 17, 18breq123d 4976 . . . 4 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
2015, 19sbie 2498 . . 3 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
217, 11, 20vtoclbg 3511 . 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 1522  [wsb 2042  wcel 2081  Vcvv 3437  [wsbc 3706  csb 3811  c0 4211   class class class wbr 4962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963
This theorem is referenced by:  sbcbr  5017  sbcbr12g  5018  csbcnvgALT  5641  sbcfung  6249  csbfv12  6581  relowlpssretop  34195
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