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Theorem elintd 38271
 Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
elintd.1 𝑥𝜑
elintd.2 (𝜑𝐴𝑉)
elintd.3 ((𝜑𝑥𝐵) → 𝐴𝑥)
Assertion
Ref Expression
elintd (𝜑𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elintd
StepHypRef Expression
1 elintd.1 . . 3 𝑥𝜑
2 elintd.3 . . . 4 ((𝜑𝑥𝐵) → 𝐴𝑥)
32ex 449 . . 3 (𝜑 → (𝑥𝐵𝐴𝑥))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝑥)
5 elintd.2 . . 3 (𝜑𝐴𝑉)
6 elintg 4418 . . 3 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
75, 6syl 17 . 2 (𝜑 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
84, 7mpbird 246 1 (𝜑𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  ∩ 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:  ssuniint  38276  elintdv  38277
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