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Theorem ss2abdv 3403
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 1641 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 ss2ab 3398 . 2  |-  ( { x  |  ps }  C_ 
{ x  |  ch } 
<-> 
A. x ( ps 
->  ch ) )
42, 3sylibr 204 1  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   {cab 2416    C_ wss 3307
This theorem is referenced by:  ssopab2  4467  opabbrex  6104  ssoprab2  6116  ss2ixp  7061  fiss  7415  tcss  7667  tcel  7668  infmap2  8082  cfub  8113  cflm  8114  cflecard  8117  cncmet  19258  plyss  20101  ofrn2  24036  sigaclci  24498  subfacp1lem6  24854  itg2addnclem  26197  sdclem1  26379  istotbnd3  26412  sstotbnd  26416  aomclem4  27064  hbtlem4  27240  hbtlem3  27241  rngunsnply  27288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-in 3314  df-ss 3321
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