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Theorem elintdv 38773
 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 1840 . 2 𝑥𝜑
2 elintdv.1 . 2 (𝜑𝐴𝑉)
3 elintdv.2 . 2 ((𝜑𝑥𝐵) → 𝐴𝑥)
41, 2, 3elintd 38767 1 (𝜑𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1987  ∩ cint 4447 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 2913  df-v 3192  df-int 4448 This theorem is referenced by: (None)
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