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Theorem intmin4 4278
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 4265 . . . 4  |-  ( A 
C_  |^| { x  | 
ph }  <->  A. x
( ph  ->  A  C_  x ) )
2 simpr 467 . . . . . . . 8  |-  ( ( A  C_  x  /\  ph )  ->  ph )
3 ancr 556 . . . . . . . 8  |-  ( (
ph  ->  A  C_  x
)  ->  ( ph  ->  ( A  C_  x  /\  ph ) ) )
42, 3impbid2 209 . . . . . . 7  |-  ( (
ph  ->  A  C_  x
)  ->  ( ( A  C_  x  /\  ph ) 
<-> 
ph ) )
54imbi1d 323 . . . . . 6  |-  ( (
ph  ->  A  C_  x
)  ->  ( (
( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
65alimi 1695 . . . . 5  |-  ( A. x ( ph  ->  A 
C_  x )  ->  A. x ( ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  ( ph  ->  y  e.  x ) ) )
7 albi 1701 . . . . 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 17 . . . 4  |-  ( A. x ( ph  ->  A 
C_  x )  -> 
( A. x ( ( A  C_  x  /\  ph )  ->  y  e.  x )  <->  A. x
( ph  ->  y  e.  x ) ) )
91, 8sylbi 200 . . 3  |-  ( A 
C_  |^| { x  | 
ph }  ->  ( A. x ( ( A 
C_  x  /\  ph )  ->  y  e.  x
)  <->  A. x ( ph  ->  y  e.  x ) ) )
10 vex 3060 . . . 4  |-  y  e. 
_V
1110elintab 4259 . . 3  |-  ( y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <->  A. x
( ( A  C_  x  /\  ph )  -> 
y  e.  x ) )
1210elintab 4259 . . 3  |-  ( y  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  y  e.  x ) )
139, 11, 123bitr4g 296 . 2  |-  ( A 
C_  |^| { x  | 
ph }  ->  (
y  e.  |^| { x  |  ( A  C_  x  /\  ph ) }  <-> 
y  e.  |^| { x  |  ph } ) )
1413eqrdv 2460 1  |-  ( A 
C_  |^| { x  | 
ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^| { x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453    = wceq 1455    e. wcel 1898   {cab 2448    C_ wss 3416   |^|cint 4248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-in 3423  df-ss 3430  df-int 4249
This theorem is referenced by: (None)
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