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Theorem dfint2 4621
 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 4620 . 2 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
2 df-ral 3047 . . 3 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
32abbii 2869 . 2 {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
41, 3eqtr4i 2777 1 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1622   = wceq 1624   ∈ wcel 2131  {cab 2738  ∀wral 3042  ∩ cint 4619 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-ral 3047  df-int 4620 This theorem is referenced by:  inteq  4622  elintg  4627  nfint  4630  intss  4642  intiin  4718  dfint3  32357
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