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Theorem intmin4 3837
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 3826 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 simpr 109 . . . . . . . 8  |-  ( ( A  C_  x  /\  ph )  ->  ph )
3 ancr 319 . . . . . . . 8  |-  ( (
ph  ->  A  C_  x
)  ->  ( ph  ->  ( A  C_  x  /\  ph ) ) )
42, 3impbid2 142 . . . . . . 7  |-  ( (
ph  ->  A  C_  x
)  ->  ( ( A  C_  x  /\  ph ) 
<-> 
ph ) )
54imbi1d 230 . . . . . 6  |-  ( (
ph  ->  A  C_  x
)  ->  ( (
( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
65alimi 1435 . . . . 5  |-  ( A. x ( ph  ->  A 
C_  x )  ->  A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
7 albi 1448 . . . . 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 120 . . 3  |-  ( A 
C_  |^| { x  | 
ph }  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
10 vex 2715 . . . 4  |-  y  e. 
_V
1110elintab 3820 . . 3  |-  ( y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  -> 
y  e.  x ) )
1210elintab 3820 . . 3  |-  ( y  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  y  e.  x ) )
139, 11, 123bitr4g 222 . 2  |-  ( A 
C_  |^| { x  | 
ph }  ->  (
y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <-> 
y  e.  |^| { x  |  ph } ) )
1413eqrdv 2155 1  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1333    = wceq 1335    e. wcel 2128   {cab 2143    C_ wss 3102   |^|cint 3809
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3810
This theorem is referenced by: (None)
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