Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  inteq Unicode version

Theorem inteq 4289
 Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
Assertion
Ref Expression
inteq

Proof of Theorem inteq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3054 . . 3
21abbidv 2593 . 2
3 dfint2 4288 . 2
4 dfint2 4288 . 2
52, 3, 43eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  {cab 2442  A.wral 2807  |^|cint 4286 This theorem is referenced by:  inteqi  4290  inteqd  4291  unissint  4311  uniintsn  4324  rint0  4327  intex  4608  intnex  4609  elreldm  5232  elxp5  6745  1stval2  6817  oev2  7192  fundmen  7609  xpsnen  7621  fiint  7817  elfir  7895  inelfi  7898  fiin  7902  cardmin2  8400  isfin2-2  8720  incexclem  13648  xpnnenOLD  13943  mreintcl  14992  ismred2  15000  fiinopn  19410  cmpfii  19909  ptbasfi  20082  fbssint  20339  shintcl  26248  chintcl  26250  rankeq1o  29828  neificl  30246  heibor1lem  30305  elrfi  30626  elrfirn  30627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-int 4287
 Copyright terms: Public domain W3C validator