Theorem abssi 3172

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 3169	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2260	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3131	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 1480   {cab 2125   C_ wss 3071

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084

This theorem is referenced by:  ssab2  3181  abf  3406  intab  3800  opabss  3992  relopabi  4665  exse2  4913  tfrlem8  6215  frecabex  6295  fiprc  6709  fival  6858  nqprxx  7354  ltnqex  7357  gtnqex  7358  recexprlemell  7430  recexprlemelu  7431  recexprlempr  7440  topnex  12255