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Theorem inteq 3742
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 2601 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21abbidv 2233 . 2 (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦})
3 dfint2 3741 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
4 dfint2 3741 . 2 𝐵 = {𝑥 ∣ ∀𝑦𝐵 𝑥𝑦}
52, 3, 43eqtr4g 2173 1 (𝐴 = 𝐵 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  {cab 2101  wral 2391   cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-int 3740
This theorem is referenced by:  inteqi  3743  inteqd  3744  uniintsnr  3775  rint0  3778  intexr  4043  onintexmid  4455  elreldm  4733  elxp5  4995  1stval2  6019  fundmen  6666  xpsnen  6681  fiintim  6783  elfir  6827  fiinopn  12066  bj-intexr  12917
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