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

Theorem ssintrab 3706
Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
Assertion
Ref Expression
ssintrab  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  C_  x
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem ssintrab
StepHypRef Expression
1 df-rab 2368 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21inteqi 3687 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
32sseq2i 3049 . 2  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) } )
4 impexp 259 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  C_  x )  <->  ( x  e.  B  ->  ( ph  ->  A  C_  x )
) )
54albii 1404 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
6 ssintab 3700 . . 3  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  C_  x )
)
7 df-ral 2364 . . 3  |-  ( A. x  e.  B  ( ph  ->  A  C_  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  C_  x
) ) )
85, 6, 73bitr4i 210 . 2  |-  ( A 
C_  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x  e.  B  ( ph  ->  A  C_  x )
)
93, 8bitri 182 1  |-  ( A 
C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  C_  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    e. wcel 1438   {cab 2074   A.wral 2359   {crab 2363    C_ wss 2997   |^|cint 3683
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-in 3003  df-ss 3010  df-int 3684
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator