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  3556  intab  3983  opabss  4179  relopabi  4885  exse2  5141  mpoexw  6422  tfrlem8  6562  frecabex  6642  fiprc  7070  fival  7270  nqprxx  7877  ltnqex  7880  gtnqex  7881  recexprlemell  7953  recexprlemelu  7954  recexprlempr  7963  4sqlem1  13111  topnex  15077  2sqlem7  16120