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Theorem intmin4 3643
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 3632 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 simpr 103 . . . . . . . 8  |-  ( ( A  C_  x  /\  ph )  ->  ph )
3 ancr 304 . . . . . . . 8  |-  ( (
ph  ->  A  C_  x
)  ->  ( ph  ->  ( A  C_  x  /\  ph ) ) )
42, 3impbid2 131 . . . . . . 7  |-  ( (
ph  ->  A  C_  x
)  ->  ( ( A  C_  x  /\  ph ) 
<-> 
ph ) )
54imbi1d 220 . . . . . 6  |-  ( (
ph  ->  A  C_  x
)  ->  ( (
( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
65alimi 1344 . . . . 5  |-  ( A. x ( ph  ->  A 
C_  x )  ->  A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
7 albi 1357 . . . . 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 114 . . 3  |-  ( A 
C_  |^| { x  | 
ph }  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
10 vex 2560 . . . 4  |-  y  e. 
_V
1110elintab 3626 . . 3  |-  ( y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  -> 
y  e.  x ) )
1210elintab 3626 . . 3  |-  ( y  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  y  e.  x ) )
139, 11, 123bitr4g 212 . 2  |-  ( A 
C_  |^| { x  | 
ph }  ->  (
y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <-> 
y  e.  |^| { x  |  ph } ) )
1413eqrdv 2038 1  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026    C_ wss 2917   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by: (None)
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