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Theorem abssi 3167
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 3164 . 2  |-  { x  |  ph }  C_  { x  |  x  e.  A }
3 abid2 2258 . 2  |-  { x  |  x  e.  A }  =  A
42, 3sseqtri 3126 1  |-  { x  |  ph }  C_  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   {cab 2123    C_ wss 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079
This theorem is referenced by:  ssab2  3176  abf  3401  intab  3795  opabss  3987  relopabi  4660  exse2  4908  tfrlem8  6208  frecabex  6288  fiprc  6702  fival  6851  nqprxx  7347  ltnqex  7350  gtnqex  7351  recexprlemell  7423  recexprlemelu  7424  recexprlempr  7433  topnex  12244
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