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Theorem abssi 3070
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
StepHypRef Expression
1 abssi.1 . . 3  |-  ( ph  ->  x  e.  A )
21ss2abi 3067 . 2  |-  { x  |  ph }  C_  { x  |  x  e.  A }
3 abid2 2200 . 2  |-  { x  |  x  e.  A }  =  A
42, 3sseqtri 3032 1  |-  { x  |  ph }  C_  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   {cab 2068    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987
This theorem is referenced by:  ssab2  3079  abf  3294  intab  3673  opabss  3850  relopabi  4491  exse2  4729  tfrlem8  5967  frecabex  6047  fiprc  6360  nqprxx  6798  ltnqex  6801  gtnqex  6802  recexprlemell  6874  recexprlemelu  6875  recexprlempr  6884
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