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Theorem ss2abdv 3569
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
Hypothesis
Ref Expression
ss2abdv.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
ss2abdv  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem ss2abdv
StepHypRef Expression
1 ss2abdv.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21alrimiv 1720 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 ss2ab 3564 . 2  |-  ( { x  |  ps }  C_ 
{ x  |  ch } 
<-> 
A. x ( ps 
->  ch ) )
42, 3sylibr 212 1  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1393   {cab 2442    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-in 3478  df-ss 3485
This theorem is referenced by:  intss  4309  ssopab2  4782  opabbrexOLD  6339  ssoprab2  6352  suppimacnvss  6927  suppimacnv  6928  ressuppss  6937  ss2ixp  7501  fiss  7902  tcss  8192  tcel  8193  infmap2  8615  cfub  8646  cflm  8647  cflecard  8650  cncmet  21887  plyss  22722  ofrn2  27628  sigaclci  28305  subfacp1lem6  28826  ss2mcls  29125  itg2addnclem  30271  sdclem1  30441  istotbnd3  30472  sstotbnd  30476  aomclem4  31207  hbtlem4  31279  hbtlem3  31280  rngunsnply  31326  iocinico  31383  clsslem  37947
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