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Theorem sbcbrg 4041
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 2958 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
2 csbeq1 3052 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3052 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
4 csbeq1 3052 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4breq123d 4001 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
6 nfcsb1v 3082 . . . 4 𝑥𝑦 / 𝑥𝐵
7 nfcsb1v 3082 . . . 4 𝑥𝑦 / 𝑥𝑅
8 nfcsb1v 3082 . . . 4 𝑥𝑦 / 𝑥𝐶
96, 7, 8nfbr 4033 . . 3 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
10 csbeq1a 3058 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3058 . . . 4 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
12 csbeq1a 3058 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1310, 11, 12breq123d 4001 . . 3 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
149, 13sbie 1784 . 2 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
151, 5, 14vtoclbg 2791 1 (𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  [wsb 1755  wcel 2141  [wsbc 2955  csb 3049   class class class wbr 3987
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by:  sbcbr12g  4042  csbcnvg  4793  sbcfung  5220  csbfv12g  5530
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