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Theorem rexin 4256
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 3979 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 624 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 468 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3088 1 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wrex 3068  cin 3962
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480  df-in 3970
This theorem is referenced by:  wefrc  5683  elidinxp  6064  imaindm  6321  bnd2  9931  subislly  23505  pcmplfin  33821
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