Theorem ssabral 3263

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 3262	. 2 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
2		df-ral 2488	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	1, 2	bitr4i 187	1 |- ( A C_ { x | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370   e. wcel 2175   {cab 2190   A.wral 2483   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-ral 2488  df-in 3171  df-ss 3178

This theorem is referenced by:  txdis1cn  14721  bj-bdfindis  15845