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Theorem elint2 4626
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
elint2.1 𝐴 ∈ V
Assertion
Ref Expression
elint2 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint2
StepHypRef Expression
1 elint2.1 . . 3 𝐴 ∈ V
21elint 4625 . 2 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
3 df-ral 3047 . 2 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
42, 3bitr4i 267 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1622  wcel 2131  wral 3042  Vcvv 3332   cint 4619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-v 3334  df-int 4620
This theorem is referenced by:  elintgOLD  4628  int0  4634  ssint  4637  intssuni  4643  iinuni  4753  trint  4912  trintssOLD  4914  onint  7152  intwun  9741  inttsk  9780  intgru  9820  subgint  17811  subrgint  18996  lssintcl  19158  toponmre  21091  alexsubALTlem3  22046  shintcli  28489  chintcli  28491  fin2so  33701  intidl  34133  mzpincl  37791  elimaint  38434  elintima  38439  intsal  41043  salgencntex  41056
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