ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inteq GIF version

Theorem inteq 3689
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 2562 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21abbidv 2205 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦})
3 dfint2 3688 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
4 dfint2 3688 . 2 𝐵 = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦}
52, 3, 43eqtr4g 2145 1 (𝐴 = 𝐵 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  {cab 2074  wral 2359   cint 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-int 3687
This theorem is referenced by:  inteqi  3690  inteqd  3691  uniintsnr  3722  rint0  3725  intexr  3984  onintexmid  4386  elreldm  4657  elxp5  4914  1stval2  5918  fundmen  6513  xpsnen  6527  fiintim  6629  fiinopn  11484  bj-intexr  11682
  Copyright terms: Public domain W3C validator