Theorem dfrab2 3531

Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
dfrab2	⊢ {x ∈ A ∣ φ} = ({x ∣ φ} ∩ A)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2		inab 3523	. . 3 ⊢ ({x ∣ x ∈ A} ∩ {x ∣ φ}) = {x ∣ (x ∈ A ∧ φ)}
3		abid2 2471	. . . 4 ⊢ {x ∣ x ∈ A} = A
4	3	ineq1i 3454	. . 3 ⊢ ({x ∣ x ∈ A} ∩ {x ∣ φ}) = (A ∩ {x ∣ φ})
5	2, 4	eqtr3i 2375	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} = (A ∩ {x ∣ φ})
6		incom 3449	. 2 ⊢ (A ∩ {x ∣ φ}) = ({x ∣ φ} ∩ A)
7	1, 5, 6	3eqtri 2377	1 ⊢ {x ∈ A ∣ φ} = ({x ∣ φ} ∩ A)

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  phialllem1  4617  frds  5936  nenpw1pwlem1  6085  nmembers1lem1  6269