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Theorem kmlem10 7801
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 7800 . 2  |-  A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
3 vex 2804 . . . . 5  |-  x  e. 
_V
43abrexex 5779 . . . 4  |-  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  { t } ) ) }  e.  _V
51, 4eqeltri 2366 . . 3  |-  A  e. 
_V
6 raleq 2749 . . . . 5  |-  ( h  =  A  ->  ( A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) )  <->  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) ) )
76raleqbi1dv 2757 . . . 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 2749 . . . . 5  |-  ( h  =  A  ->  ( A. z  e.  h  ph  <->  A. z  e.  A  ph ) )
98exbidv 1616 . . . 4  |-  ( h  =  A  ->  ( E. y A. z  e.  h  ph  <->  E. y A. z  e.  A  ph ) )
107, 9imbi12d 311 . . 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 2887 . 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 16 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 set class
Syntax hints:    -> wi 4   A.wal 1530   E.wex 1531    = wceq 1632   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164   (/)c0 3468   {csn 3653   U.cuni 3843
This theorem is referenced by:  kmlem13  7804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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