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Theorem csbov1g 5937
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov1g (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbov1g
StepHypRef Expression
1 csbov12g 5936 . 2 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶))
2 csbconstg 3086 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32oveq2d 5913 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐹𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
41, 3eqtrd 2222 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐹𝐶) = (𝐴 / 𝑥𝐵𝐹𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  csb 3072  (class class class)co 5897
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900
This theorem is referenced by:  modfsummodlemstep  11500  fprodmodd  11684
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