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Theorem elinti 4450
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 4448 . . 3 (𝐴 𝐵 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
2 eleq2 2687 . . . 4 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
32rspccv 3292 . . 3 (∀𝑥𝐵 𝐴𝑥 → (𝐶𝐵𝐴𝐶))
41, 3syl6bi 243 . 2 (𝐴 𝐵 → (𝐴 𝐵 → (𝐶𝐵𝐴𝐶)))
54pm2.43i 52 1 (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wral 2907   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-int 4441
This theorem is referenced by:  inttsk  9540  subgint  17539  subrgint  18723  lssintcl  18883  ufinffr  21643  shintcli  28037  insiga  29981  intsal  39855
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