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Theorem elint 3595
 Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)
Hypothesis
Ref Expression
elint.1 A V
Assertion
Ref Expression
elint (A Bx(x BA x))
Distinct variable groups:   x,A   x,B

Proof of Theorem elint
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elint.1 . 2 A V
2 eleq1 2082 . . . 4 (y = A → (y xA x))
32imbi2d 219 . . 3 (y = A → ((x By x) ↔ (x BA x)))
43albidv 1687 . 2 (y = A → (x(x By x) ↔ x(x BA x)))
5 df-int 3590 . 2 B = {yx(x By x)}
61, 4, 5elab2 2667 1 (A Bx(x BA x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2535  ∩ cint 3589 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-int 3590 This theorem is referenced by:  elint2  3596  elintab  3600  intss1  3604  intss  3610  intun  3620  intpr  3621  peano1  4244
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