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Theorem sbcbr123 4739
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 3478 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 ∈ V)
2 br0 4734 . . . 4 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
3 csbprc 4013 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
43breqd 4696 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
52, 4mtbiri 316 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
65con4i 113 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 ∈ V)
7 dfsbcq2 3471 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
8 csbeq1 3569 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
9 csbeq1 3569 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
10 csbeq1 3569 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
118, 9, 10breq123d 4699 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
12 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝐵
13 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝑅
14 nfcsb1v 3582 . . . . 5 𝑥𝑦 / 𝑥𝐶
1512, 13, 14nfbr 4732 . . . 4 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
16 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
17 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
18 csbeq1a 3575 . . . . 5 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1916, 17, 18breq123d 4699 . . . 4 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
2015, 19sbie 2436 . . 3 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
217, 11, 20vtoclbg 3298 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
221, 6, 21pm5.21nii 367 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  [wsb 1937  wcel 2030  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  sbcbr  4740  sbcbr12g  4741  csbcnvgALT  5339  sbcfung  5950  csbfv12  6269  relowlpssretop  33342
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