Theorem abssi 3177

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 3174	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2261	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3136	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 1481   {cab 2126   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3082  df-ss 3089

This theorem is referenced by:  ssab2  3186  abf  3411  intab  3808  opabss  4000  relopabi  4673  exse2  4921  tfrlem8  6223  frecabex  6303  fiprc  6717  fival  6866  nqprxx  7378  ltnqex  7381  gtnqex  7382  recexprlemell  7454  recexprlemelu  7455  recexprlempr  7464  topnex  12294