New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  eluni2 GIF version

Theorem eluni2 3895
 Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
eluni2 (A Bx B A x)
Distinct variable groups:   x,A   x,B

Proof of Theorem eluni2
StepHypRef Expression
1 exancom 1586 . 2 (x(A x x B) ↔ x(x B A x))
2 eluni 3894 . 2 (A Bx(A x x B))
3 df-rex 2620 . 2 (x B A xx(x B A x))
41, 2, 33bitr4i 268 1 (A Bx B A x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃wrex 2615  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-uni 3892 This theorem is referenced by:  uni0b  3916  intssuni  3948  iuncom4  3976  dfuni3  4315  eqpw1uni  4330  elfin  4420  cnvuni  4895  chfnrn  5399
 Copyright terms: Public domain W3C validator