Theorem ssabral 3319

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 3318	. 2 |- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
2		df-ral 2533	. 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 1400   e. wcel 2209   {cab 2224   A.wral 2528   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-in 3226  df-ss 3233

This theorem is referenced by:  txdis1cn  15302  bj-bdfindis  16887