ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss2abdv Unicode version

Theorem ss2abdv 3083
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 1799 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 ss2ab 3078 . 2  |-  ( { x  |  ps }  C_ 
{ x  |  ch } 
<-> 
A. x ( ps 
->  ch ) )
42, 3sylibr 132 1  |-  ( ph  ->  { x  |  ps }  C_  { x  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285   {cab 2071    C_ wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-in 2994  df-ss 3001
This theorem is referenced by:  ssopab2  4075  iotass  4960  imadif  5056  imain  5058  opabbrex  5644  ssoprab2  5656  tfr1onlemssrecs  6052  tfrcllemssrecs  6065
  Copyright terms: Public domain W3C validator