Theorem ssab 3223

Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
ssab	|- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ssab

Step	Hyp	Ref	Expression
1		abid2 2296	. . 3 |- { x | x e. A } = A
2	1	sseq1i 3179	. 2 |- ( { x | x e. A } C_ { x | ph } <-> A C_ { x | ph } )
3		ss2ab 3221	. 2 |- ( { x | x e. A } C_ { x | ph } <-> A. x ( x e. A -> ph ) )
4	2, 3	bitr3i 186	1 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   e. wcel 2146   {cab 2161   C_ wss 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-in 3133  df-ss 3140

This theorem is referenced by:  ssabral  3224  ssrab  3231