Theorem ssab 3262

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 2325	. . 3 |- { x | x e. A } = A
2	1	sseq1i 3218	. 2 |- ( { x | x e. A } C_ { x | ph } <-> A C_ { x | ph } )
3		ss2ab 3260	. 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 1370   e. wcel 2175   {cab 2190   C_ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-in 3171  df-ss 3178

This theorem is referenced by:  ssabral  3263  ssrab  3270