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Theorem elintrab 3655
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 3654 . . 3  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( ( x  e.  B  /\  ph )  ->  A  e.  x ) )
3 impexp 254 . . . 4  |-  ( ( ( x  e.  B  /\  ph )  ->  A  e.  x )  <->  ( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
43albii 1375 . . 3  |-  ( A. x ( ( x  e.  B  /\  ph )  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
52, 4bitri 177 . 2  |-  ( A  e.  |^| { x  |  ( x  e.  B  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( ph  ->  A  e.  x ) ) )
6 df-rab 2332 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
76inteqi 3647 . . 3  |-  |^| { x  e.  B  |  ph }  =  |^| { x  |  ( x  e.  B  /\  ph ) }
87eleq2i 2120 . 2  |-  ( A  e.  |^| { x  e.  B  |  ph }  <->  A  e.  |^| { x  |  ( x  e.  B  /\  ph ) } )
9 df-ral 2328 . 2  |-  ( A. x  e.  B  ( ph  ->  A  e.  x
)  <->  A. x ( x  e.  B  ->  ( ph  ->  A  e.  x
) ) )
105, 8, 93bitr4i 205 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 101    <-> wb 102   A.wal 1257    e. wcel 1409   {cab 2042   A.wral 2323   {crab 2327   _Vcvv 2574   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-int 3644
This theorem is referenced by:  elintrabg  3656  intmin  3663  bj-indint  10442
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