Theorem abssi 3254

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 3251	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2314	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3213	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2164   {cab 2179   C_ wss 3153

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 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166

This theorem is referenced by:  ssab2  3263  abf  3490  intab  3899  opabss  4093  relopabi  4781  exse2  5031  mpoexw  6257  tfrlem8  6362  frecabex  6442  fiprc  6860  fival  7019  nqprxx  7596  ltnqex  7599  gtnqex  7600  recexprlemell  7672  recexprlemelu  7673  recexprlempr  7682  4sqlem1  12513  topnex  14225  2sqlem7  15146