Theorem ssabral 3337

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 C_ {x | ph} <-> A.x e. A ph)

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem ssabral

Step	Hyp	Ref	Expression
1		ssab 3336	. 2 |- (A C_ {x | ph} <-> A.x(x e. A -> ph))
2		df-ral 2619	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
3	1, 2	bitr4i 243	1 |- (A C_ {x | ph} <-> A.x e. A ph)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2614   C_ wss 3257

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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)