Theorem abssi 3317

Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Hypothesis

Ref	Expression
abssi.1	|- ( ph -> x e. A )

Assertion

Ref	Expression
abssi	|- { x | ph } C_ A

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem abssi

Step	Hyp	Ref	Expression
1		abssi.1	. . 3 |- ( ph -> x e. A )
2	1	ss2abi 3314	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2357	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3276	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2205   {cab 2220   C_ wss 3214

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227

This theorem is referenced by:  ssab2  3326  abf  3559  intab  3984  opabss  4180  relopabi  4887  exse2  5143  mpoexw  6424  tfrlem8  6564  frecabex  6644  fiprc  7072  fival  7272  nqprxx  7879  ltnqex  7882  gtnqex  7883  recexprlemell  7955  recexprlemelu  7956  recexprlempr  7965  4sqlem1  13117  topnex  15082  2sqlem7  16125