Theorem abssi 3245

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 3242	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2310	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3204	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2160   {cab 2175   C_ wss 3144

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157

This theorem is referenced by:  ssab2  3254  abf  3481  intab  3888  opabss  4082  relopabi  4770  exse2  5020  mpoexw  6237  tfrlem8  6342  frecabex  6422  fiprc  6840  fival  6998  nqprxx  7574  ltnqex  7577  gtnqex  7578  recexprlemell  7650  recexprlemelu  7651  recexprlempr  7660  4sqlem1  12419  topnex  14038  2sqlem7  14921