Theorem abssi 3299

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 3296	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2350	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3258	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2200   {cab 2215   C_ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210

This theorem is referenced by:  ssab2  3308  abf  3535  intab  3952  opabss  4148  relopabi  4847  exse2  5102  mpoexw  6359  tfrlem8  6464  frecabex  6544  fiprc  6968  fival  7137  nqprxx  7733  ltnqex  7736  gtnqex  7737  recexprlemell  7809  recexprlemelu  7810  recexprlempr  7819  4sqlem1  12911  topnex  14760  2sqlem7  15800