Theorem rabss 3344

Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
rabss	⊢ ({x ∈ A ∣ φ} ⊆ B ↔ ∀x ∈ A (φ → x ∈ B))

Distinct variable group:   x,B

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rabss

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2	1	sseq1i 3296	. 2 ⊢ ({x ∈ A ∣ φ} ⊆ B ↔ {x ∣ (x ∈ A ∧ φ)} ⊆ B)
3		abss 3336	. 2 ⊢ ({x ∣ (x ∈ A ∧ φ)} ⊆ B ↔ ∀x((x ∈ A ∧ φ) → x ∈ B))
4		impexp 433	. . . 4 ⊢ (((x ∈ A ∧ φ) → x ∈ B) ↔ (x ∈ A → (φ → x ∈ B)))
5	4	albii 1566	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → x ∈ B) ↔ ∀x(x ∈ A → (φ → x ∈ B)))
6		df-ral 2620	. . 3 ⊢ (∀x ∈ A (φ → x ∈ B) ↔ ∀x(x ∈ A → (φ → x ∈ B)))
7	5, 6	bitr4i 243	. 2 ⊢ (∀x((x ∈ A ∧ φ) → x ∈ B) ↔ ∀x ∈ A (φ → x ∈ B))
8	2, 3, 7	3bitri 262	1 ⊢ ({x ∈ A ∣ φ} ⊆ B ↔ ∀x ∈ A (φ → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2615  {crab 2619   ⊆ wss 3258

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-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  rabssdv  3347