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Theorem dfint2 3780
 Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
dfint2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfint2
StepHypRef Expression
1 df-int 3779 . 2 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
2 df-ral 2422 . . 3 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
32abbii 2256 . 2 {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
41, 3eqtr4i 2164 1 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∩ cint 3778 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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-ral 2422  df-int 3779 This theorem is referenced by:  inteq  3781  nfint  3788  intiin  3874
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