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Theorem intmin4 3711
Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
Assertion
Ref Expression
intmin4  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem intmin4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssintab 3700 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 simpr 108 . . . . . . . 8  |-  ( ( A  C_  x  /\  ph )  ->  ph )
3 ancr 314 . . . . . . . 8  |-  ( (
ph  ->  A  C_  x
)  ->  ( ph  ->  ( A  C_  x  /\  ph ) ) )
42, 3impbid2 141 . . . . . . 7  |-  ( (
ph  ->  A  C_  x
)  ->  ( ( A  C_  x  /\  ph ) 
<-> 
ph ) )
54imbi1d 229 . . . . . 6  |-  ( (
ph  ->  A  C_  x
)  ->  ( (
( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
65alimi 1389 . . . . 5  |-  ( A. x ( ph  ->  A 
C_  x )  ->  A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
7 albi 1402 . . . . 5  |-  ( A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) )  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
86, 7syl 14 . . . 4  |-  ( A. x ( ph  ->  A 
C_  x )  -> 
( A. x ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  A. x
( ph  ->  y  e.  x ) ) )
91, 8sylbi 119 . . 3  |-  ( A 
C_  |^| { x  | 
ph }  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
10 vex 2622 . . . 4  |-  y  e. 
_V
1110elintab 3694 . . 3  |-  ( y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  -> 
y  e.  x ) )
1210elintab 3694 . . 3  |-  ( y  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  y  e.  x ) )
139, 11, 123bitr4g 221 . 2  |-  ( A 
C_  |^| { x  | 
ph }  ->  (
y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <-> 
y  e.  |^| { x  |  ph } ) )
1413eqrdv 2086 1  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074    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-v 2621  df-in 3003  df-ss 3010  df-int 3684
This theorem is referenced by: (None)
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