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Theorem rexin 4232
Description: Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018.)
Assertion
Ref Expression
rexin (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))

Proof of Theorem rexin
StepHypRef Expression
1 elin 3957 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 624 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 469 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 274 . 2 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3089 1 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2106  wrex 3069  cin 3940
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3472  df-in 3948
This theorem is referenced by:  wefrc  5660  elidinxp  6030  imaindm  6284  bnd2  9867  subislly  22909  pcmplfin  32655
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