Theorem ssabral 3090

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 3089	. 2 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
2		df-ral 2364	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	1, 2	bitr4i 185	1 |- ( A C_ { x | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   e. wcel 1438   {cab 2074   A.wral 2359   C_ wss 2997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3003  df-ss 3010

This theorem is referenced by:  bj-bdfindis  11488