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Theorem elintd 39559
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 2986 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝑥)
5 elintd.2 . . 3 (𝜑𝐴𝑉)
6 elintg 4515 . . 3 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
75, 6syl 17 . 2 (𝜑 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
84, 7mpbird 247 1 (𝜑𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wnf 1748  wcel 2030  wral 2941   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-int 4508
This theorem is referenced by:  ssuniint  39564  elintdv  39565
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