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Theorem nfco 5197
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 5037 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2750 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2750 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 4623 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2750 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 4623 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1815 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2139 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 4644 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2748 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 382  wex 1694  wnfc 2737   class class class wbr 4577  {copab 4636  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-co 5037
This theorem is referenced by:  nffun  5812  nftpos  7251  cnmpt11  21218  cnmpt21  21226  poimirlem16  32391  poimirlem19  32394  csbcog  36756  choicefi  38183  cncficcgt0  38571  volioofmpt  38684  volicofmpt  38687  stoweidlem31  38721  stoweidlem59  38749
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