Theorem abssi 3070

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 3067	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2200	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3032	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 1434   {cab 2068   C_ wss 2974

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987

This theorem is referenced by:  ssab2  3079  abf  3294  intab  3673  opabss  3850  relopabi  4491  exse2  4729  tfrlem8  5967  frecabex  6047  fiprc  6360  nqprxx  6798  ltnqex  6801  gtnqex  6802  recexprlemell  6874  recexprlemelu  6875  recexprlempr  6884