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Theorem dfint2 4851
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 4850 . 2 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
2 df-ral 3131 . . 3 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
32abbii 2886 . 2 {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
41, 3eqtr4i 2847 1 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wcel 2115  {cab 2799  wral 3126   cint 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-ral 3131  df-int 4850
This theorem is referenced by:  inteq  4852  elintg  4857  nfint  4859  intss  4870  intiin  4956  dfint3  33420
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