Theorem abssi 3228

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 3225	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2296	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3187	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2146   {cab 2161   C_ wss 3127

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-in 3133  df-ss 3140

This theorem is referenced by:  ssab2  3237  abf  3464  intab  3869  opabss  4062  relopabi  4746  exse2  4995  mpoexw  6204  tfrlem8  6309  frecabex  6389  fiprc  6805  fival  6959  nqprxx  7520  ltnqex  7523  gtnqex  7524  recexprlemell  7596  recexprlemelu  7597  recexprlempr  7606  4sqlem1  12351  topnex  13137  2sqlem7  14008