Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexdifpr Structured version   Visualization version   GIF version

Theorem rexdifpr 40315
 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 4152 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴𝑥𝐵𝑥𝐶))
2 3anass 1035 . . . . 5 ((𝑥𝐴𝑥𝐵𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
31, 2bitri 263 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
43anbi1i 727 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑))
5 anass 679 . . . 4 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)))
6 df-3an 1033 . . . . . 6 ((𝑥𝐵𝑥𝐶𝜑) ↔ ((𝑥𝐵𝑥𝐶) ∧ 𝜑))
76bicomi 213 . . . . 5 (((𝑥𝐵𝑥𝐶) ∧ 𝜑) ↔ (𝑥𝐵𝑥𝐶𝜑))
87anbi2i 726 . . . 4 ((𝑥𝐴 ∧ ((𝑥𝐵𝑥𝐶) ∧ 𝜑)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
95, 8bitri 263 . . 3 (((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
104, 9bitri 263 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶𝜑)))
1110rexbii2 3021 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  usgr2pth0  40971
 Copyright terms: Public domain W3C validator