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Theorem rexin 4098
 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 4053 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 614 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 461 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 267 . 2 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3186 1 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387   ∈ wcel 2048  ∃wrex 3083   ∩ cin 3824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-v 3411  df-in 3832 This theorem is referenced by:  wefrc  5394  elidinxp  5750  bnd2  9108  subislly  21783  pcmplfin  30725  imaindm  32482
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