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Theorem sbcbrg 3952
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
sbcbrg (𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))

Proof of Theorem sbcbrg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2885 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
2 csbeq1 2978 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 2978 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
4 csbeq1 2978 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4breq123d 3913 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
6 nfcsb1v 3005 . . . 4 𝑥𝑦 / 𝑥𝐵
7 nfcsb1v 3005 . . . 4 𝑥𝑦 / 𝑥𝑅
8 nfcsb1v 3005 . . . 4 𝑥𝑦 / 𝑥𝐶
96, 7, 8nfbr 3944 . . 3 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
10 csbeq1a 2983 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 2983 . . . 4 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
12 csbeq1a 2983 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1310, 11, 12breq123d 3913 . . 3 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
149, 13sbie 1749 . 2 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
151, 5, 14vtoclbg 2721 1 (𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  [wsb 1720  [wsbc 2882  csb 2975   class class class wbr 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900
This theorem is referenced by:  sbcbr12g  3953  csbcnvg  4693  sbcfung  5117  csbfv12g  5425
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