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Theorem elinintrab 40193
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 3483 . . . 4 𝑥 ∈ V
21inex2 5208 . . 3 (𝐵𝑥) ∈ V
3 inss1 4190 . . 3 (𝐵𝑥) ⊆ 𝐵
42, 3elmapintrab 40192 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)))))
5 elin 3935 . . . . . . . 8 (𝐴 ∈ (𝐵𝑥) ↔ (𝐴𝐵𝐴𝑥))
65imbi2i 339 . . . . . . 7 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ (𝜑 → (𝐴𝐵𝐴𝑥)))
7 jcab 521 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
86, 7bitri 278 . . . . . 6 ((𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
98albii 1821 . . . . 5 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)))
10 19.26 1872 . . . . . 6 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
11 19.23v 1944 . . . . . . 7 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
1211anbi1i 626 . . . . . 6 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1310, 12bitri 278 . . . . 5 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
149, 13bitri 278 . . . 4 (∀𝑥(𝜑𝐴 ∈ (𝐵𝑥)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1514anbi2i 625 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
16 anabs5 662 . . 3 (((∃𝑥𝜑𝐴𝐵) ∧ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
1715, 16bitri 278 . 2 (((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴 ∈ (𝐵𝑥))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥)))
184, 17syl6bb 290 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  {crab 3137  cin 3918  𝒫 cpw 4522   cint 4862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-int 4863
This theorem is referenced by:  inintabss  40194  inintabd  40195
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