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Theorem ss2abdv 3246
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 1617 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 ss2ab 3241 . 2  |-  ( { x  |  ps }  C_ 
{ x  |  ch } 
<-> 
A. x ( ps 
->  ch ) )
42, 3sylibr 203 1  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   {cab 2269    C_ wss 3152
This theorem is referenced by:  ssopab2  4290  ssoprab2  5904  ss2ixp  6829  fiss  7177  tcss  7429  tcel  7430  infmap2  7844  cfub  7875  cflm  7876  cflecard  7879  cncmet  18744  plyss  19581  ofrn2  23207  sigaclci  23493  subfacp1lem6  23716  sdclem1  26453  istotbnd3  26495  sstotbnd  26499  aomclem4  27154  hbtlem4  27330  hbtlem3  27331  rngunsnply  27378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166
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