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Theorem csbov123g 5912
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 (𝐴𝐷𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))

Proof of Theorem csbov123g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3060 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐵𝐹𝐶) = 𝐴 / 𝑥(𝐵𝐹𝐶))
2 csbeq1 3060 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3060 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
4 csbeq1 3060 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
52, 3, 4oveq123d 5895 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
61, 5eqeq12d 2192 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)))
7 vex 2740 . . 3 𝑦 ∈ V
8 nfcsb1v 3090 . . . 4 𝑥𝑦 / 𝑥𝐵
9 nfcsb1v 3090 . . . 4 𝑥𝑦 / 𝑥𝐹
10 nfcsb1v 3090 . . . 4 𝑥𝑦 / 𝑥𝐶
118, 9, 10nfov 5904 . . 3 𝑥(𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
12 csbeq1a 3066 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
13 csbeq1a 3066 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
14 csbeq1a 3066 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
1512, 13, 14oveq123d 5895 . . 3 (𝑥 = 𝑦 → (𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶))
167, 11, 15csbief 3101 . 2 𝑦 / 𝑥(𝐵𝐹𝐶) = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐹𝑦 / 𝑥𝐶)
176, 16vtoclg 2797 1 (𝐴𝐷𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  csb 3057  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by:  csbov12g  5913
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