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Theorem elinti 4420
 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 → (𝐶𝐵𝐴𝐶))

Proof of Theorem elinti
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elintg 4418 . . 3 (𝐴 𝐵 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
2 eleq2 2677 . . . 4 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
32rspccv 3279 . . 3 (∀𝑥𝐵 𝐴𝑥 → (𝐶𝐵𝐴𝐶))
41, 3syl6bi 242 . 2 (𝐴 𝐵 → (𝐴 𝐵 → (𝐶𝐵𝐴𝐶)))
54pm2.43i 50 1 (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-int 4411 This theorem is referenced by:  inttsk  9475  subgint  17441  subrgint  18625  lssintcl  18785  ufinffr  21543  shintcli  27572  insiga  29527  intsal  39224
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