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Theorem kmlem10 8620
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 8619 . 2  |-  A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
3 vex 3060 . . . . 5  |-  x  e. 
_V
43abrexex 6799 . . . 4  |-  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  { t } ) ) }  e.  _V
51, 4eqeltri 2536 . . 3  |-  A  e. 
_V
6 raleq 2999 . . . . 5  |-  ( h  =  A  ->  ( A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) )  <->  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) ) )
76raleqbi1dv 3007 . . . 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 2999 . . . . 5  |-  ( h  =  A  ->  ( A. z  e.  h  ph  <->  A. z  e.  A  ph ) )
98exbidv 1779 . . . 4  |-  ( h  =  A  ->  ( E. y A. z  e.  h  ph  <->  E. y A. z  e.  A  ph ) )
107, 9imbi12d 326 . . 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 3152 . 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 20 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 1453    = wceq 1455   E.wex 1674   {cab 2448    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057    \ cdif 3413    i^i cin 3415   (/)c0 3743   {csn 3980   U.cuni 4212
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-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613
This theorem is referenced by:  kmlem13  8623
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