Theorem ssabral 3213

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 3212	. 2 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
2		df-ral 2449	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	1, 2	bitr4i 186	1 |- ( A C_ { x | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   e. wcel 2136   {cab 2151   A.wral 2444   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129

This theorem is referenced by:  txdis1cn  12918  bj-bdfindis  13829