MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexdifpr Structured version   Visualization version   GIF version

Theorem rexdifpr 4588
Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
Assertion
Ref Expression
rexdifpr (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))

Proof of Theorem rexdifpr
StepHypRef Expression
1 eldifpr 4587 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴𝑥𝐵𝑥𝐶))
2 3anass 1087 . . . . 5 ((𝑥𝐴𝑥𝐵𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
31, 2bitri 276 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
43anbi1i 623 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑))
5 anass 469 . . . 4 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
6 df-3an 1081 . . . . . 6 ((𝑥𝐵𝑥𝐶𝜑) ↔ ((𝑥𝐵𝑥𝐶) ∧ 𝜑))
76bicomi 225 . . . . 5 (((𝑥𝐵𝑥𝐶) ∧ 𝜑) ↔ (𝑥𝐵𝑥𝐶𝜑))
87anbi2i 622 . . . 4 ((𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
95, 8bitri 276 . . 3 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
104, 9bitri 276 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
1110rexbii2 3242 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079  wcel 2105  wne 3013  wrex 3136  cdif 3930  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  usgr2pth0  27473
  Copyright terms: Public domain W3C validator