Theorem abssi 3276

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 3273	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2328	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3235	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2178   {cab 2193   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187

This theorem is referenced by:  ssab2  3285  abf  3512  intab  3928  opabss  4124  relopabi  4821  exse2  5075  mpoexw  6322  tfrlem8  6427  frecabex  6507  fiprc  6931  fival  7098  nqprxx  7694  ltnqex  7697  gtnqex  7698  recexprlemell  7770  recexprlemelu  7771  recexprlempr  7780  4sqlem1  12826  topnex  14673  2sqlem7  15713