Theorem abssi 3232

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 3229	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2298	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3191	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2148   {cab 2163   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144

This theorem is referenced by:  ssab2  3241  abf  3468  intab  3875  opabss  4069  relopabi  4754  exse2  5004  mpoexw  6216  tfrlem8  6321  frecabex  6401  fiprc  6817  fival  6971  nqprxx  7547  ltnqex  7550  gtnqex  7551  recexprlemell  7623  recexprlemelu  7624  recexprlempr  7633  4sqlem1  12388  topnex  13671  2sqlem7  14553