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Theorem kmlem10 8528
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1  |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  {
t } ) ) }
Assertion
Ref Expression
kmlem10  |-  ( A. h ( A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  ->  E. y A. z  e.  A  ph )
Distinct variable groups:    x, y,
z, w, u, t, h    y, A, z, w, h    ph, h
Allowed substitution hints:    ph( x, y, z, w, u, t)    A( x, u, t)

Proof of Theorem kmlem10
StepHypRef Expression
1 kmlem9.1 . . 3  |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  {
t } ) ) }
21kmlem9 8527 . 2  |-  A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
3 vex 3109 . . . . 5  |-  x  e. 
_V
43abrexex 6748 . . . 4  |-  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  { t } ) ) }  e.  _V
51, 4eqeltri 2544 . . 3  |-  A  e. 
_V
6 raleq 3051 . . . . 5  |-  ( h  =  A  ->  ( A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) )  <->  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) ) )
76raleqbi1dv 3059 . . . 4  |-  ( h  =  A  ->  ( A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) )  <->  A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) ) )
8 raleq 3051 . . . . 5  |-  ( h  =  A  ->  ( A. z  e.  h  ph  <->  A. z  e.  A  ph ) )
98exbidv 1685 . . . 4  |-  ( h  =  A  ->  ( E. y A. z  e.  h  ph  <->  E. y A. z  e.  A  ph ) )
107, 9imbi12d 320 . . 3  |-  ( h  =  A  ->  (
( A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  <->  ( A. z  e.  A  A. w  e.  A  (
z  =/=  w  -> 
( z  i^i  w
)  =  (/) )  ->  E. y A. z  e.  A  ph ) ) )
115, 10spcv 3197 . 2  |-  ( A. h ( A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  -> 
( A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  A  ph ) )
122, 11mpi 17 1  |-  ( A. h ( A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  ->  E. y A. z  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1372    = wceq 1374   E.wex 1591   {cab 2445    =/= wne 2655   A.wral 2807   E.wrex 2808   _Vcvv 3106    \ cdif 3466    i^i cin 3468   (/)c0 3778   {csn 4020   U.cuni 4238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by:  kmlem13  8531
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