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Theorem abssi 3015
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 3012 . 2  |-  { x  |  ph }  C_  { x  |  x  e.  A }
3 abid2 2158 . 2  |-  { x  |  x  e.  A }  =  A
42, 3sseqtri 2977 1  |-  { x  |  ph }  C_  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1393   {cab 2026    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931
This theorem is referenced by:  ssab2  3024  abf  3260  intab  3644  opabss  3821  relopabi  4463  exse2  4699  tfrlem8  5934  frecabex  5984  fiprc  6292  nqprxx  6642  ltnqex  6645  gtnqex  6646  recexprlemell  6718  recexprlemelu  6719  recexprlempr  6728
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