Theorem rabid2 2789

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
rabid2	⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		abeq2 2459	. . 3 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
2		pm4.71 611	. . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ∧ φ)))
3	2	albii 1566	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
4	1, 3	bitr4i 243	. 2 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A → φ))
5		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
6	5	eqeq2i 2363	. 2 ⊢ (A = {x ∈ A ∣ φ} ↔ A = {x ∣ (x ∈ A ∧ φ)})
7		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
8	4, 6, 7	3bitr4i 268	1 ⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  {crab 2619

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-rab 2624

This theorem is referenced by:  rabxm  3574  iinrab2  4030  riinrab  4042  opeq  4620  dmmptg  5685  fmpt  5693