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Theorem elintdv 38277
Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
elintdv.1 (𝜑𝐴𝑉)
elintdv.2 ((𝜑𝑥𝐵) → 𝐴𝑥)
Assertion
Ref Expression
elintdv (𝜑𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elintdv
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 elintdv.1 . 2 (𝜑𝐴𝑉)
3 elintdv.2 . 2 ((𝜑𝑥𝐵) → 𝐴𝑥)
41, 2, 3elintd 38271 1 (𝜑𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977   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: (None)
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