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Theorem sbcbrg 3804
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 (A 𝐷 → ([A / x]B𝑅𝐶A / xBA / x𝑅A / x𝐶))

Proof of Theorem sbcbrg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 (y = A → ([y / x]B𝑅𝐶[A / x]B𝑅𝐶))
2 csbeq1 2849 . . 3 (y = Ay / xB = A / xB)
3 csbeq1 2849 . . 3 (y = Ay / x𝑅 = A / x𝑅)
4 csbeq1 2849 . . 3 (y = Ay / x𝐶 = A / x𝐶)
52, 3, 4breq123d 3769 . 2 (y = A → (y / xBy / x𝑅y / x𝐶A / xBA / x𝑅A / x𝐶))
6 nfcsb1v 2876 . . . 4 xy / xB
7 nfcsb1v 2876 . . . 4 xy / x𝑅
8 nfcsb1v 2876 . . . 4 xy / x𝐶
96, 7, 8nfbr 3799 . . 3 xy / xBy / x𝑅y / x𝐶
10 csbeq1a 2854 . . . 4 (x = yB = y / xB)
11 csbeq1a 2854 . . . 4 (x = y𝑅 = y / x𝑅)
12 csbeq1a 2854 . . . 4 (x = y𝐶 = y / x𝐶)
1310, 11, 12breq123d 3769 . . 3 (x = y → (B𝑅𝐶y / xBy / x𝑅y / x𝐶))
149, 13sbie 1671 . 2 ([y / x]B𝑅𝐶y / xBy / x𝑅y / x𝐶)
151, 5, 14vtoclbg 2608 1 (A 𝐷 → ([A / x]B𝑅𝐶A / xBA / x𝑅A / x𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  [wsb 1642  [wsbc 2758  csb 2846   class class class wbr 3755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756
This theorem is referenced by:  sbcbr12g  3805  csbcnvg  4462  sbcfung  4868  csbfv12g  5152
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