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Theorem abssdv 3093
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssdv.1  |-  ( ph  ->  ( ps  ->  x  e.  A ) )
Assertion
Ref Expression
abssdv  |-  ( ph  ->  { x  |  ps }  C_  A )
Distinct variable groups:    ph, x    x, A
Allowed substitution hint:    ps( x)

Proof of Theorem abssdv
StepHypRef Expression
1 abssdv.1 . . 3  |-  ( ph  ->  ( ps  ->  x  e.  A ) )
21alrimiv 1802 . 2  |-  ( ph  ->  A. x ( ps 
->  x  e.  A
) )
3 abss 3088 . 2  |-  ( { x  |  ps }  C_  A  <->  A. x ( ps 
->  x  e.  A
) )
42, 3sylibr 132 1  |-  ( ph  ->  { x  |  ps }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    e. wcel 1438   {cab 2074    C_ wss 2997
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3003  df-ss 3010
This theorem is referenced by:  fmpt  5433  tfrlemibacc  6073  tfrlemibfn  6075  tfr1onlembacc  6089  tfr1onlembfn  6091  tfrcllembacc  6102  tfrcllembfn  6104  eroprf  6365  genipv  7047  hashfacen  10206
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