MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfco Structured version   Visualization version   GIF version

Theorem nfco 5857
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 5678 . 2 (𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
2 nfcv 2902 . . . . . 6 𝑥𝑦
3 nfco.2 . . . . . 6 𝑥𝐵
4 nfcv 2902 . . . . . 6 𝑥𝑤
52, 3, 4nfbr 5188 . . . . 5 𝑥 𝑦𝐵𝑤
6 nfco.1 . . . . . 6 𝑥𝐴
7 nfcv 2902 . . . . . 6 𝑥𝑧
84, 6, 7nfbr 5188 . . . . 5 𝑥 𝑤𝐴𝑧
95, 8nfan 1902 . . . 4 𝑥(𝑦𝐵𝑤𝑤𝐴𝑧)
109nfex 2317 . . 3 𝑥𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)
1110nfopab 5210 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤𝑤𝐴𝑧)}
121, 11nfcxfr 2900 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wex 1781  wnfc 2882   class class class wbr 5141  {copab 5203  ccom 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-co 5678
This theorem is referenced by:  csbcog  6285  nffun  6560  nftpos  8228  nfwrecs  8283  cnmpt11  23096  cnmpt21  23104  poimirlem16  36308  poimirlem19  36311  choicefi  43670  cncficcgt0  44377  volioofmpt  44483  volicofmpt  44486  stoweidlem31  44520  stoweidlem59  44548
  Copyright terms: Public domain W3C validator