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Theorem elinintrab 37702
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 3198 . . . 4 𝑥 ∈ V
21inex2 4791 . . 3 (𝐵𝑥) ∈ V
3 inss1 3825 . . 3 (𝐵𝑥) ⊆ 𝐵
42, 3elmapintrab 37701 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)))))
5 elin 3788 . . . . . . . 8 (𝐴 ∈ (𝐵𝑥) ↔ (𝐴𝐵𝐴𝑥))
65imbi2i 326 . . . . . . 7 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ (𝜑 → (𝐴𝐵𝐴𝑥)))
7 jcab 906 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
86, 7bitri 264 . . . . . 6 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
98albii 1745 . . . . 5 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
10 19.26 1796 . . . . . 6 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
11 19.23v 1900 . . . . . . 7 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
1211anbi1i 730 . . . . . 6 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1310, 12bitri 264 . . . . 5 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
149, 13bitri 264 . . . 4 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1514anbi2i 729 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
16 anabs5 850 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1715, 16bitri 264 . 2 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
184, 17syl6bb 276 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  {crab 2913  cin 3566  𝒫 cpw 4149   cint 4466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-int 4467
This theorem is referenced by:  inintabss  37703  inintabd  37704
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