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Theorem nfco 4810
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 4653 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2332 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2332 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 4064 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2332 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 4064 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1576 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 1648 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 4086 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2329 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wex 1503  wnfc 2319   class class class wbr 4018  {copab 4078  ccom 4648
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-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-co 4653
This theorem is referenced by:  nffun  5258  nftpos  6303  cnmpt11  14235  cnmpt21  14243
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