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Theorem sbcbrg 4685
 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 / xBRC[A / x]B[A / x]R[A / x]C))

Proof of Theorem sbcbrg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (y = A → ([y / x]BRC ↔ [̣A / xBRC))
2 csbeq1 3139 . . 3 (y = A[y / x]B = [A / x]B)
3 csbeq1 3139 . . 3 (y = A[y / x]R = [A / x]R)
4 csbeq1 3139 . . 3 (y = A[y / x]C = [A / x]C)
52, 3, 4breq123d 4653 . 2 (y = A → ([y / x]B[y / x]R[y / x]C[A / x]B[A / x]R[A / x]C))
6 nfcsb1v 3168 . . . 4 x[y / x]B
7 nfcsb1v 3168 . . . 4 x[y / x]R
8 nfcsb1v 3168 . . . 4 x[y / x]C
96, 7, 8nfbr 4683 . . 3 x[y / x]B[y / x]R[y / x]C
10 csbeq1a 3144 . . . 4 (x = yB = [y / x]B)
11 csbeq1a 3144 . . . 4 (x = yR = [y / x]R)
12 csbeq1a 3144 . . . 4 (x = yC = [y / x]C)
1310, 11, 12breq123d 4653 . . 3 (x = y → (BRC[y / x]B[y / x]R[y / x]C))
149, 13sbie 2038 . 2 ([y / x]BRC[y / x]B[y / x]R[y / x]C)
151, 5, 14vtoclbg 2915 1 (A D → ([̣A / xBRC[A / x]B[A / x]R[A / x]C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046  [csb 3136   class class class wbr 4639 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640 This theorem is referenced by:  sbcbr12g  4686  csbfv12g  5336
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