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Theorem elinti 4959
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 4958 . . 3 (𝐴 𝐵 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
2 eleq2 2821 . . . 4 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
32rspccv 3609 . . 3 (∀𝑥𝐵 𝐴𝑥 → (𝐶𝐵𝐴𝐶))
41, 3biimtrdi 252 . 2 (𝐴 𝐵 → (𝐴 𝐵 → (𝐶𝐵𝐴𝐶)))
54pm2.43i 52 1 (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3060   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-int 4951
This theorem is referenced by:  inttsk  10775  subgint  19073  subrngint  20456  subrgint  20493  lssintcl  20807  ufinffr  23754  shintcli  31017  intlidl  32978  insiga  33601  intsal  45508
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