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Theorem elinintrab 36902
Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.)
Assertion
Ref Expression
elinintrab (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Distinct variable groups:   𝜑,𝑤   𝑥,𝑤,𝐴   𝑤,𝐵,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥,𝑤)

Proof of Theorem elinintrab
StepHypRef Expression
1 vex 3176 . . . 4 𝑥 ∈ V
21inex2 4728 . . 3 (𝐵𝑥) ∈ V
3 inss1 3795 . . 3 (𝐵𝑥) ⊆ 𝐵
42, 3elmapintrab 36901 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)))))
5 elin 3758 . . . . . . . 8 (𝐴 ∈ (𝐵𝑥) ↔ (𝐴𝐵𝐴𝑥))
65imbi2i 325 . . . . . . 7 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ (𝜑 → (𝐴𝐵𝐴𝑥)))
7 jcab 903 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
86, 7bitri 263 . . . . . 6 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
98albii 1737 . . . . 5 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
10 19.26 1786 . . . . . 6 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
11 19.23v 1889 . . . . . . 7 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
1211anbi1i 727 . . . . . 6 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1310, 12bitri 263 . . . . 5 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
149, 13bitri 263 . . . 4 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1514anbi2i 726 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
16 anabs5 847 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1715, 16bitri 263 . 2 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
184, 17syl6bb 275 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  wcel 1977  {crab 2900  cin 3539  𝒫 cpw 4108   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  ax-sep 4709
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  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-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-int 4411
This theorem is referenced by:  inintabss  36903  inintabd  36904
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