Theorem abssi 3303

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 3300	. 2 |- { x | ph } C_ { x | x e. A }
3		abid2 2353	. 2 |- { x | x e. A } = A
4	2, 3	sseqtri 3262	1 |- { x | ph } C_ A

Syntax hints:   -> wi 4   e. wcel 2202   {cab 2217   C_ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214

This theorem is referenced by:  ssab2  3312  abf  3540  intab  3962  opabss  4158  relopabi  4861  exse2  5117  mpoexw  6387  tfrlem8  6527  frecabex  6607  fiprc  7033  fival  7229  nqprxx  7826  ltnqex  7829  gtnqex  7830  recexprlemell  7902  recexprlemelu  7903  recexprlempr  7912  4sqlem1  13041  topnex  14897  2sqlem7  15940