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Theorem sbcbrg 3841
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 2790 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶[𝐴 / 𝑥]𝐵𝑅𝐶))
2 csbeq1 2883 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 2883 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝑅 = 𝐴 / 𝑥𝑅)
4 csbeq1 2883 . . 3 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4breq123d 3806 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
6 nfcsb1v 2910 . . . 4 𝑥𝑦 / 𝑥𝐵
7 nfcsb1v 2910 . . . 4 𝑥𝑦 / 𝑥𝑅
8 nfcsb1v 2910 . . . 4 𝑥𝑦 / 𝑥𝐶
96, 7, 8nfbr 3836 . . 3 𝑥𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶
10 csbeq1a 2888 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 2888 . . . 4 (𝑥 = 𝑦𝑅 = 𝑦 / 𝑥𝑅)
12 csbeq1a 2888 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1310, 11, 12breq123d 3806 . . 3 (𝑥 = 𝑦 → (𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶))
149, 13sbie 1690 . 2 ([𝑦 / 𝑥]𝐵𝑅𝐶𝑦 / 𝑥𝐵𝑦 / 𝑥𝑅𝑦 / 𝑥𝐶)
151, 5, 14vtoclbg 2631 1 (𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  [wsb 1661  [wsbc 2787  csb 2880   class class class wbr 3792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793
This theorem is referenced by:  sbcbr12g  3842  csbcnvg  4547  sbcfung  4953  csbfv12g  5237
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