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Theorem nfco 5822
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 5643 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2904 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2904 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 5153 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2904 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 5153 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1903 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2318 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 5175 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2902 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wnfc 2884   class class class wbr 5106  {copab 5168  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-co 5643
This theorem is referenced by:  csbcog  6250  nffun  6525  nftpos  8193  nfwrecs  8248  cnmpt11  23030  cnmpt21  23038  poimirlem16  36140  poimirlem19  36143  choicefi  43508  cncficcgt0  44215  volioofmpt  44321  volicofmpt  44324  stoweidlem31  44358  stoweidlem59  44386
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