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

Theorem elintrabg 3651
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elintrabg  |-  ( A  e.  V  ->  ( 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)    V( x)

Proof of Theorem elintrabg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2142 . 2  |-  ( y  =  A  ->  (
y  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  e.  B  |  ph }
) )
2 eleq1 2142 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 228 . . 3  |-  ( y  =  A  ->  (
( ph  ->  y  e.  x )  <->  ( ph  ->  A  e.  x ) ) )
43ralbidv 2369 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  ( ph  ->  y  e.  x )  <->  A. x  e.  B  ( ph  ->  A  e.  x ) ) )
5 vex 2605 . . 3  |-  y  e. 
_V
65elintrab 3650 . 2  |-  ( y  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  y  e.  x
) )
71, 4, 6vtoclbg 2660 1  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  (
ph  ->  A  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353   |^|cint 3638
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-int 3639
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator