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Theorem abssdv 3221
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 1867 . 2  |-  ( ph  ->  A. x ( ps 
->  x  e.  A
) )
3 abss 3216 . 2  |-  ( { x  |  ps }  C_  A  <->  A. x ( ps 
->  x  e.  A
) )
42, 3sylibr 133 1  |-  ( ph  ->  { x  |  ps }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    e. wcel 2141   {cab 2156    C_ wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134
This theorem is referenced by:  fmpt  5644  tfrlemibacc  6303  tfrlemibfn  6305  tfr1onlembacc  6319  tfr1onlembfn  6321  tfrcllembacc  6332  tfrcllembfn  6334  eroprf  6604  genipv  7464  hashfacen  10764  metrest  13265
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