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Theorem csbov123g 5485
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
csbov123g (A 𝐷A / x(B𝐹𝐶) = (A / xBA / x𝐹A / x𝐶))

Proof of Theorem csbov123g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2849 . . 3 (y = Ay / x(B𝐹𝐶) = A / x(B𝐹𝐶))
2 csbeq1 2849 . . . 4 (y = Ay / x𝐹 = A / x𝐹)
3 csbeq1 2849 . . . 4 (y = Ay / xB = A / xB)
4 csbeq1 2849 . . . 4 (y = Ay / x𝐶 = A / x𝐶)
52, 3, 4oveq123d 5476 . . 3 (y = A → (y / xBy / x𝐹y / x𝐶) = (A / xBA / x𝐹A / x𝐶))
61, 5eqeq12d 2051 . 2 (y = A → (y / x(B𝐹𝐶) = (y / xBy / x𝐹y / x𝐶) ↔ A / x(B𝐹𝐶) = (A / xBA / x𝐹A / x𝐶)))
7 vex 2554 . . 3 y V
8 nfcsb1v 2876 . . . 4 xy / xB
9 nfcsb1v 2876 . . . 4 xy / x𝐹
10 nfcsb1v 2876 . . . 4 xy / x𝐶
118, 9, 10nfov 5478 . . 3 x(y / xBy / x𝐹y / x𝐶)
12 csbeq1a 2854 . . . 4 (x = y𝐹 = y / x𝐹)
13 csbeq1a 2854 . . . 4 (x = yB = y / xB)
14 csbeq1a 2854 . . . 4 (x = y𝐶 = y / x𝐶)
1512, 13, 14oveq123d 5476 . . 3 (x = y → (B𝐹𝐶) = (y / xBy / x𝐹y / x𝐶))
167, 11, 15csbief 2885 . 2 y / x(B𝐹𝐶) = (y / xBy / x𝐹y / x𝐶)
176, 16vtoclg 2607 1 (A 𝐷A / x(B𝐹𝐶) = (A / xBA / x𝐹A / x𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  csb 2846  (class class class)co 5455
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-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  csbov12g  5486
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