Theorem dfint2 4477

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 4476	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
2		df-ral 2917	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))
3	2	abbii 2739	. 2 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
4	1, 3	eqtr4i 2647	1 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}

Syntax hints:   → wi 4  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∩ cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-int 4476

This theorem is referenced by:  inteq  4478  elintg  4483  nfint  4486  intss  4498  intiin  4574  dfint3  32059