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Theorem ss2abdv 3418
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 1642 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 ss2ab 3413 . 2  |-  ( { x  |  ps }  C_ 
{ x  |  ch } 
<-> 
A. x ( ps 
->  ch ) )
42, 3sylibr 205 1  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   {cab 2424    C_ wss 3322
This theorem is referenced by:  ssopab2  4483  opabbrex  6121  ssoprab2  6133  ss2ixp  7078  fiss  7432  tcss  7686  tcel  7687  infmap2  8103  cfub  8134  cflm  8135  cflecard  8138  cncmet  19280  plyss  20123  ofrn2  24058  sigaclci  24520  subfacp1lem6  24876  itg2addnclem  26270  sdclem1  26461  istotbnd3  26494  sstotbnd  26498  aomclem4  27146  hbtlem4  27321  hbtlem3  27322  rngunsnply  27369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336
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