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Theorem elinti 4642
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 4641 . . 3 (𝐴 𝐵 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
2 eleq2 2833 . . . 4 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
32rspccv 3458 . . 3 (∀𝑥𝐵 𝐴𝑥 → (𝐶𝐵𝐴𝐶))
41, 3syl6bi 244 . 2 (𝐴 𝐵 → (𝐴 𝐵 → (𝐶𝐵𝐴𝐶)))
54pm2.43i 52 1 (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2155  wral 3055   cint 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-int 4634
This theorem is referenced by:  inttsk  9849  subgint  17884  subrgint  19071  lssintcl  19236  ufinffr  22012  shintcli  28579  insiga  30582  intsal  41117
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