Theorem rabss2 3350

Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rabss2	⊢ (A ⊆ B → {x ∈ A ∣ φ} ⊆ {x ∈ B ∣ φ})

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem rabss2

Step	Hyp	Ref	Expression
1		pm3.45 807	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ∧ φ) → (x ∈ B ∧ φ)))
2	1	alimi 1559	. . 3 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ φ)))
3		dfss2 3263	. . 3 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
4		ss2ab 3335	. . 3 ⊢ ({x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ (x ∈ B ∧ φ)} ↔ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ φ)))
5	2, 3, 4	3imtr4i 257	. 2 ⊢ (A ⊆ B → {x ∣ (x ∈ A ∧ φ)} ⊆ {x ∣ (x ∈ B ∧ φ)})
6		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
7		df-rab 2624	. 2 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)}
8	5, 6, 7	3sstr4g 3313	1 ⊢ (A ⊆ B → {x ∈ A ∣ φ} ⊆ {x ∈ B ∣ φ})

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  {crab 2619   ⊆ wss 3258

This proof depends on 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 proof 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  df-ss 3260

This theorem is used by: (None)