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Theorem elinintrab 43567
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 3482 . . . 4 𝑥 ∈ V
21inex2 5324 . . 3 (𝐵𝑥) ∈ V
3 inss1 4245 . . 3 (𝐵𝑥) ⊆ 𝐵
42, 3elmapintrab 43566 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)))))
5 elin 3979 . . . . . . . 8 (𝐴 ∈ (𝐵𝑥) ↔ (𝐴𝐵𝐴𝑥))
65imbi2i 336 . . . . . . 7 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ (𝜑 → (𝐴𝐵𝐴𝑥)))
7 jcab 517 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
86, 7bitri 275 . . . . . 6 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
98albii 1816 . . . . 5 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
10 19.26 1868 . . . . . 6 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
11 19.23v 1940 . . . . . . 7 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
1211anbi1i 624 . . . . . 6 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1310, 12bitri 275 . . . . 5 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
149, 13bitri 275 . . . 4 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1514anbi2i 623 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
16 anabs5 663 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1715, 16bitri 275 . 2 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
184, 17bitrdi 287 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {crab 3433  cin 3962  𝒫 cpw 4605   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-int 4952
This theorem is referenced by:  inintabss  43568  inintabd  43569
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