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Theorem nfco 5458
Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
Hypotheses
Ref Expression
nfco.1 𝑥𝐴
nfco.2 𝑥𝐵
Assertion
Ref Expression
nfco 𝑥(𝐴𝐵)

Proof of Theorem nfco
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-co 5288 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2907 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2907 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 4858 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2907 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 4858 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1998 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2327 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 4879 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2905 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384  wex 1874  wnfc 2894   class class class wbr 4811  {copab 4873  ccom 5283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-co 5288
This theorem is referenced by:  nffun  6093  nftpos  7594  cnmpt11  21760  cnmpt21  21768  poimirlem16  33870  poimirlem19  33873  csbcog  38640  choicefi  40061  cncficcgt0  40763  volioofmpt  40872  volicofmpt  40875  stoweidlem31  40909  stoweidlem59  40937
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