Theorem abssi 3268

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 3265	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2326	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3227	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2176   {cab 2191   C_ wss 3166

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179

This theorem is referenced by:  ssab2  3277  abf  3504  intab  3914  opabss  4109  relopabi  4804  exse2  5057  mpoexw  6301  tfrlem8  6406  frecabex  6486  fiprc  6909  fival  7074  nqprxx  7661  ltnqex  7664  gtnqex  7665  recexprlemell  7737  recexprlemelu  7738  recexprlempr  7747  4sqlem1  12744  topnex  14591  2sqlem7  15631