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Theorem nfco 5863
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 5682 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2899 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2899 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 5190 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2899 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 5190 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1895 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2313 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 5212 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2897 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1774  wnfc 2879   class class class wbr 5143  {copab 5205  ccom 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-co 5682
This theorem is referenced by:  csbcog  6296  nffun  6571  nftpos  8261  nfwrecs  8316  cnmpt11  23561  cnmpt21  23569  poimirlem16  37104  poimirlem19  37107  choicefi  44564  cncficcgt0  45267  volioofmpt  45373  volicofmpt  45376  stoweidlem31  45410  stoweidlem59  45438
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