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Theorem elintg 3934
 Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
elintg (A V → (A Bx B A x))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   V(x)

Proof of Theorem elintg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . 2 (y = A → (y BA B))
2 eleq1 2413 . . 3 (y = A → (y xA x))
32ralbidv 2634 . 2 (y = A → (x B y xx B A x))
4 vex 2862 . . 3 y V
54elint2 3933 . 2 (y Bx B y x)
61, 3, 5vtoclbg 2915 1 (A V → (A Bx B A x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ral 2619  df-v 2861  df-int 3927 This theorem is referenced by:  elinti  3935  elrint  3967
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