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Theorem intabs 4363
Description: Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intabs.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
intabs.2  |-  ( x  =  |^| { y  |  ps }  ->  (
ph 
<->  ch ) )
intabs.3  |-  ( |^| { y  |  ps }  C_  A  /\  ch )
Assertion
Ref Expression
intabs  |-  |^| { x  |  ( x  C_  A  /\  ph ) }  =  |^| { x  |  ph }
Distinct variable groups:    x, y    x, A    ph, y    ps, x    ch, x
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    A( y)

Proof of Theorem intabs
StepHypRef Expression
1 sseq1 3371 . . . . . 6  |-  ( x  =  |^| { y  |  ps }  ->  ( x  C_  A  <->  |^| { y  |  ps }  C_  A ) )
2 intabs.2 . . . . . 6  |-  ( x  =  |^| { y  |  ps }  ->  (
ph 
<->  ch ) )
31, 2anbi12d 693 . . . . 5  |-  ( x  =  |^| { y  |  ps }  ->  ( ( x  C_  A  /\  ph )  <->  ( |^| { y  |  ps }  C_  A  /\  ch )
) )
4 intabs.3 . . . . 5  |-  ( |^| { y  |  ps }  C_  A  /\  ch )
53, 4intmin3 4080 . . . 4  |-  ( |^| { y  |  ps }  e.  _V  ->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps } )
6 intnex 4359 . . . . 5  |-  ( -. 
|^| { y  |  ps }  e.  _V  <->  |^| { y  |  ps }  =  _V )
7 ssv 3370 . . . . . 6  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  _V
8 sseq2 3372 . . . . . 6  |-  ( |^| { y  |  ps }  =  _V  ->  ( |^| { x  |  ( x 
C_  A  /\  ph ) }  C_  |^| { y  |  ps }  <->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  _V ) )
97, 8mpbiri 226 . . . . 5  |-  ( |^| { y  |  ps }  =  _V  ->  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps } )
106, 9sylbi 189 . . . 4  |-  ( -. 
|^| { y  |  ps }  e.  _V  ->  |^|
{ x  |  ( x  C_  A  /\  ph ) }  C_  |^| { y  |  ps } )
115, 10pm2.61i 159 . . 3  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { y  |  ps }
12 intabs.1 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1312cbvabv 2557 . . . 4  |-  { x  |  ph }  =  {
y  |  ps }
1413inteqi 4056 . . 3  |-  |^| { x  |  ph }  =  |^| { y  |  ps }
1511, 14sseqtr4i 3383 . 2  |-  |^| { x  |  ( x  C_  A  /\  ph ) } 
C_  |^| { x  | 
ph }
16 simpr 449 . . . 4  |-  ( ( x  C_  A  /\  ph )  ->  ph )
1716ss2abi 3417 . . 3  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  | 
ph }
18 intss 4073 . . 3  |-  ( { x  |  ( x 
C_  A  /\  ph ) }  C_  { x  |  ph }  ->  |^| { x  |  ph }  C_  |^| { x  |  ( x  C_  A  /\  ph ) } )
1917, 18ax-mp 8 . 2  |-  |^| { x  |  ph }  C_  |^| { x  |  ( x  C_  A  /\  ph ) }
2015, 19eqssi 3366 1  |-  |^| { x  |  ( x  C_  A  /\  ph ) }  =  |^| { x  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958    C_ wss 3322   |^|cint 4052
This theorem is referenced by:  dfnn3  10016
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-int 4053
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