Theorem dfint2 3872

Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
dfint2	⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfint2

Step	Hyp	Ref	Expression
1		df-int 3871	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
2		df-ral 2477	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
3	2	abbii 2309	. 2 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
4	1, 3	eqtr4i 2217	1 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∩ cint 3870

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-ral 2477  df-int 3871

This theorem is referenced by:  inteq  3873  nfint  3880  intiin  3967