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Theorem csbcog 37422
Description: Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
Assertion
Ref Expression
csbcog (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem csbcog
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3517 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵𝐶))
2 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
3 csbeq1 3517 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
42, 3coeq12d 5246 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
51, 4eqeq12d 2636 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
6 vex 3189 . . 3 𝑦 ∈ V
7 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐵
8 nfcsb1v 3530 . . . 4 𝑥𝑦 / 𝑥𝐶
97, 8nfco 5247 . . 3 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
10 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
11 csbeq1a 3523 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1210, 11coeq12d 5246 . . 3 (𝑥 = 𝑦 → (𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶))
136, 9, 12csbief 3539 . 2 𝑦 / 𝑥(𝐵𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐶)
145, 13vtoclg 3252 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  csb 3514  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-co 5083
This theorem is referenced by:  brtrclfv2  37500
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