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Theorem dfint2 3846
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 3845 . 2 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
2 df-ral 2460 . . 3 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
32abbii 2293 . 2 {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
41, 3eqtr4i 2201 1 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wral 2455   cint 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-int 3845
This theorem is referenced by:  inteq  3847  nfint  3854  intiin  3940
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