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Theorem nfin 3782
 Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfin.1 𝑥𝐴
nfin.2 𝑥𝐵
Assertion
Ref Expression
nfin 𝑥(𝐴𝐵)

Proof of Theorem nfin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfin5 3548 . 2 (𝐴𝐵) = {𝑦𝐴𝑦𝐵}
2 nfin.2 . . . 4 𝑥𝐵
32nfcri 2745 . . 3 𝑥 𝑦𝐵
4 nfin.1 . . 3 𝑥𝐴
53, 4nfrab 3100 . 2 𝑥{𝑦𝐴𝑦𝐵}
61, 5nfcxfr 2749 1 𝑥(𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Ⅎwnfc 2738  {crab 2900   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547 This theorem is referenced by:  csbin  3962  iunxdif3  4542  disjxun  4581  nfres  5319  nfpred  5602  cp  8637  tskwe  8659  iuncon  21041  ptclsg  21228  restmetu  22185  limciun  23464  disjunsn  28789  ordtconlem1  29298  esum2d  29482  finminlem  31482  mbfposadd  32627  csbingOLD  38076  iunconlem2  38193  inn0f  38268  disjrnmpt2  38370  disjinfi  38375  fsumiunss  38642  stoweidlem57  38950  fourierdlem80  39079  sge0iunmptlemre  39308  iundjiun  39353  pimiooltgt  39598  smflim  39663
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