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Theorem sbcbr123 4909
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 3654 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 ∈ V)
2 br0 4904 . . . 4 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
3 csbprc 4189 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
43breqd 4866 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
52, 4mtbiri 318 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
65con4i 114 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 ∈ V)
7 dfsbcq2 3647 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
8 csbeq1 3742 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
9 csbeq1 3742 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
10 csbeq1 3742 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
118, 9, 10breq123d 4869 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
12 nfcsb1v 3755 . . . . 5 𝑥𝑦 / 𝑥𝐵
13 nfcsb1v 3755 . . . . 5 𝑥𝑦 / 𝑥𝑅
14 nfcsb1v 3755 . . . . 5 𝑥𝑦 / 𝑥𝐶
1512, 13, 14nfbr 4902 . . . 4 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
16 csbeq1a 3748 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
17 csbeq1a 3748 . . . . 5 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
18 csbeq1a 3748 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1916, 17, 18breq123d 4869 . . . 4 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
2015, 19sbie 2569 . . 3 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
217, 11, 20vtoclbg 3471 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
221, 6, 21pm5.21nii 369 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1637  [wsb 2061  wcel 2157  Vcvv 3402  [wsbc 3644  csb 3739  c0 4127   class class class wbr 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856
This theorem is referenced by:  sbcbr  4910  sbcbr12g  4911  csbcnvgALT  5519  sbcfung  6132  csbfv12  6458  relowlpssretop  33534
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