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

Theorem inteq 4407
Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
Assertion
Ref Expression
inteq (𝐴 = 𝐵 𝐴 = 𝐵)

Proof of Theorem inteq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3114 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21abbidv 2727 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦})
3 dfint2 4406 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
4 dfint2 4406 . 2 𝐵 = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦}
52, 3, 43eqtr4g 2668 1 (𝐴 = 𝐵 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  {cab 2595  wral 2895   cint 4404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-int 4405
This theorem is referenced by:  inteqi  4408  inteqd  4409  unissint  4430  uniintsn  4443  rint0  4446  intex  4742  intnex  4743  elreldm  5258  elxp5  6981  1stval2  7053  oev2  7467  fundmen  7893  xpsnen  7906  fiint  8099  elfir  8181  inelfi  8184  fiin  8188  cardmin2  8684  isfin2-2  9001  incexclem  14353  mreintcl  16024  ismred2  16032  fiinopn  20473  cmpfii  20964  ptbasfi  21136  fbssint  21394  shintcl  27379  chintcl  27381  inelpisys  29350  rankeq1o  31254  neificl  32522  heibor1lem  32581  elrfi  36078  elrfirn  36079
  Copyright terms: Public domain W3C validator