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Theorem elinti 3935
 Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elinti (A B → (C BA C))

Proof of Theorem elinti
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elintg 3934 . . 3 (A B → (A Bx B A x))
2 eleq2 2414 . . . 4 (x = C → (A xA C))
32rspccv 2952 . . 3 (x B A x → (C BA C))
41, 3syl6bi 219 . 2 (A B → (A B → (C BA C)))
54pm2.43i 43 1 (A B → (C BA C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∩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-ral 2619  df-v 2861  df-int 3927 This theorem is referenced by: (None)
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