Theorem abssi 3097

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 3094	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2209	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3059	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 1439   {cab 2075   C_ wss 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-in 3006  df-ss 3013

This theorem is referenced by:  ssab2  3106  abf  3330  intab  3723  opabss  3908  relopabi  4576  exse2  4819  tfrlem8  6097  frecabex  6177  fiprc  6586  nqprxx  7159  ltnqex  7162  gtnqex  7163  recexprlemell  7235  recexprlemelu  7236  recexprlempr  7245  topnex  11840