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

Theorem elintrab 3778
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
Hypothesis
Ref Expression
inteqab.1  |-  A  e. 
_V
Assertion
Ref Expression
elintrab  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elintrab
StepHypRef Expression
1 inteqab.1 . . . 4  |-  A  e. 
_V
21elintab 3777 . . 3  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  e.  x ) )
3 impexp 261 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  e.  x )  <->  ( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
43albii 1446 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
52, 4bitri 183 . 2  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
6 df-rab 2423 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
76inteqi 3770 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
87eleq2i 2204 . 2  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  |  ( x  e.  B  /\  ph ) } )
9 df-ral 2419 . 2  |-  ( A. x  e.  B  ( ph  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
105, 8, 93bitr4i 211 1  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    e. wcel 1480   {cab 2123   A.wral 2414   {crab 2418   _Vcvv 2681   |^|cint 3766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-int 3767
This theorem is referenced by:  elintrabg  3779  intmin  3786  bj-indint  13118
  Copyright terms: Public domain W3C validator