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Theorem elint 3932
 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 2413 . . . 4 (y = A → (y xA x))
32imbi2d 307 . . 3 (y = A → ((x By x) ↔ (x BA x)))
43albidv 1625 . 2 (y = A → (x(x By x) ↔ x(x BA x)))
5 df-int 3927 . 2 B = {y x(x By x)}
61, 4, 5elab2 2988 1 (A Bx(x BA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-int 3927 This theorem is referenced by:  elint2  3933  elintab  3937  intss1  3941  intss  3947  intun  3958  intpr  3959
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