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Theorem nfco 5629
 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 5459 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2951 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2951 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 5015 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2951 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 5015 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1885 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2308 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 5036 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2949 1 𝑥(𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396  ∃wex 1765  Ⅎwnfc 2935   class class class wbr 4968  {copab 5030   ∘ ccom 5454 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-co 5459 This theorem is referenced by:  nffun  6255  nftpos  7785  cnmpt11  21959  cnmpt21  21967  poimirlem16  34460  poimirlem19  34463  csbcog  39500  choicefi  41024  cncficcgt0  41734  volioofmpt  41843  volicofmpt  41846  stoweidlem31  41880  stoweidlem59  41908
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