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Theorem dfint2 3646
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 3645 . 2 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
2 df-ral 2354 . . 3 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
32abbii 2195 . 2 {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
41, 3eqtr4i 2105 1 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wcel 1434  {cab 2068  wral 2349   cint 3644
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-ral 2354  df-int 3645
This theorem is referenced by:  inteq  3647  nfint  3654  intiin  3740
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