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Theorem elinintrab 36795
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 3080 . . . 4 𝑥 ∈ V
21inex2 4627 . . 3 (𝐵𝑥) ∈ V
3 inss1 3698 . . 3 (𝐵𝑥) ⊆ 𝐵
42, 3elmapintrab 36794 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)))))
5 elin 3662 . . . . . . . 8 (𝐴 ∈ (𝐵𝑥) ↔ (𝐴𝐵𝐴𝑥))
65imbi2i 324 . . . . . . 7 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ (𝜑 → (𝐴𝐵𝐴𝑥)))
7 jcab 902 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
86, 7bitri 262 . . . . . 6 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
98albii 1722 . . . . 5 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
10 19.26 1767 . . . . . 6 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
11 19.23v 1852 . . . . . . 7 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
1211anbi1i 726 . . . . . 6 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1310, 12bitri 262 . . . . 5 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
149, 13bitri 262 . . . 4 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1514anbi2i 725 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
16 anabs5 846 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1715, 16bitri 262 . 2 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
184, 17syl6bb 274 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1938  {crab 2804  cin 3443  𝒫 cpw 4011   cint 4308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rab 2809  df-v 3079  df-in 3451  df-ss 3458  df-pw 4013  df-int 4309
This theorem is referenced by:  inintabss  36796  inintabd  36797
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