Theorem ssabral 3338

Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Assertion

Ref	Expression
ssabral	⊢ (A ⊆ {x ∣ φ} ↔ ∀x ∈ A φ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem ssabral

Step	Hyp	Ref	Expression
1		ssab 3337	. 2 ⊢ (A ⊆ {x ∣ φ} ↔ ∀x(x ∈ A → φ))
2		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
3	1, 2	bitr4i 243	1 ⊢ (A ⊆ {x ∣ φ} ↔ ∀x ∈ A φ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2615   ⊆ 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-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is used by: (None)