Theorem ssabral 3065

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 3064	. 2 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
2		df-ral 2353	. 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 1282   e. wcel 1433   {cab 2067   A.wral 2348   C_ wss 2973

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986

This theorem is referenced by:  bj-bdfindis  10742