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Theorem csbov2g 5818
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov2g (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbov2g
StepHypRef Expression
1 csbov12g 5816 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3019 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
32oveq1d 5795 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
41, 3eqtrd 2173 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐵𝐹𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  csb 3006  (class class class)co 5780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3078  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-iota 5094  df-fv 5137  df-ov 5783
This theorem is referenced by:  csbnegg  7982  fsummulc2  11247
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