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Theorem kmlem10 8003
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 8002 . 2  |-  A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
3 vex 2927 . . . . 5  |-  x  e. 
_V
43abrexex 5950 . . . 4  |-  { u  |  E. t  e.  x  u  =  ( t  \  U. ( x  \  { t } ) ) }  e.  _V
51, 4eqeltri 2482 . . 3  |-  A  e. 
_V
6 raleq 2872 . . . . 5  |-  ( h  =  A  ->  ( A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) )  <->  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) ) )
76raleqbi1dv 2880 . . . 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 2872 . . . . 5  |-  ( h  =  A  ->  ( A. z  e.  h  ph  <->  A. z  e.  A  ph ) )
98exbidv 1633 . . . 4  |-  ( h  =  A  ->  ( E. y A. z  e.  h  ph  <->  E. y A. z  e.  A  ph ) )
107, 9imbi12d 312 . . 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 3010 . 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 set class
Syntax hints:    -> wi 4   A.wal 1546   E.wex 1547    = wceq 1649   {cab 2398    =/= wne 2575   A.wral 2674   E.wrex 2675   _Vcvv 2924    \ cdif 3285    i^i cin 3287   (/)c0 3596   {csn 3782   U.cuni 3983
This theorem is referenced by:  kmlem13  8006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429
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