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

Theorem inteq 3660
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 2555 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21abbidv 2200 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦})
3 dfint2 3659 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
4 dfint2 3659 . 2 𝐵 = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦}
52, 3, 43eqtr4g 2140 1 (𝐴 = 𝐵 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  {cab 2069  wral 2353   cint 3657
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-int 3658
This theorem is referenced by:  inteqi  3661  inteqd  3662  uniintsnr  3693  rint0  3696  intexr  3946  onintexmid  4344  elreldm  4609  elxp5  4860  1stval2  5835  fundmen  6376  xpsnen  6388  bj-intexr  10991
  Copyright terms: Public domain W3C validator