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Theorem intmin4 4080
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 4068 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 simpr 449 . . . . . . . 8  |-  ( ( A  C_  x  /\  ph )  ->  ph )
3 ancr 534 . . . . . . . 8  |-  ( (
ph  ->  A  C_  x
)  ->  ( ph  ->  ( A  C_  x  /\  ph ) ) )
42, 3impbid2 197 . . . . . . 7  |-  ( (
ph  ->  A  C_  x
)  ->  ( ( A  C_  x  /\  ph ) 
<-> 
ph ) )
54imbi1d 310 . . . . . 6  |-  ( (
ph  ->  A  C_  x
)  ->  ( (
( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
65alimi 1569 . . . . 5  |-  ( A. x ( ph  ->  A 
C_  x )  ->  A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
7 albi 1574 . . . . 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 16 . . . 4  |-  ( A. x ( ph  ->  A 
C_  x )  -> 
( A. x ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  A. x
( ph  ->  y  e.  x ) ) )
91, 8sylbi 189 . . 3  |-  ( A 
C_  |^| { x  | 
ph }  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
10 vex 2960 . . . 4  |-  y  e. 
_V
1110elintab 4062 . . 3  |-  ( y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  -> 
y  e.  x ) )
1210elintab 4062 . . 3  |-  ( y  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  y  e.  x ) )
139, 11, 123bitr4g 281 . 2  |-  ( A 
C_  |^| { x  | 
ph }  ->  (
y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <-> 
y  e.  |^| { x  |  ph } ) )
1413eqrdv 2435 1  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2423    C_ wss 3321   |^|cint 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-in 3328  df-ss 3335  df-int 4052
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