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Theorem ceqsex4v 2662
Description: Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
Hypotheses
Ref Expression
ceqsex4v.1  |-  A  e. 
_V
ceqsex4v.2  |-  B  e. 
_V
ceqsex4v.3  |-  C  e. 
_V
ceqsex4v.4  |-  D  e. 
_V
ceqsex4v.7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex4v.8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
ceqsex4v.9  |-  ( z  =  C  ->  ( ch 
<->  th ) )
ceqsex4v.10  |-  ( w  =  D  ->  ( th 
<->  ta ) )
Assertion
Ref Expression
ceqsex4v  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ta )
Distinct variable groups:    x, y, z, w, A    x, B, y, z, w    x, C, y, z, w    x, D, y, z, w    ps, x    ch, y    th, z    ta, w
Allowed substitution hints:    ph( x, y, z, w)    ps( y, z, w)    ch( x, z, w)    th( x, y, w)    ta( x, y, z)

Proof of Theorem ceqsex4v
StepHypRef Expression
1 19.42vv 1836 . . . 4  |-  ( E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D  /\  ph )
)  <->  ( ( x  =  A  /\  y  =  B )  /\  E. z E. w ( z  =  C  /\  w  =  D  /\  ph )
) )
2 3anass 928 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B )  /\  (
( z  =  C  /\  w  =  D )  /\  ph )
) )
3 df-3an 926 . . . . . . 7  |-  ( ( z  =  C  /\  w  =  D  /\  ph )  <->  ( ( z  =  C  /\  w  =  D )  /\  ph ) )
43anbi2i 445 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D  /\  ph )
)  <->  ( ( x  =  A  /\  y  =  B )  /\  (
( z  =  C  /\  w  =  D )  /\  ph )
) )
52, 4bitr4i 185 . . . . 5  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D  /\  ph ) ) )
652exbii 1542 . . . 4  |-  ( E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  E. z E. w
( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D  /\  ph ) ) )
7 df-3an 926 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  E. z E. w ( z  =  C  /\  w  =  D  /\  ph ) )  <->  ( (
x  =  A  /\  y  =  B )  /\  E. z E. w
( z  =  C  /\  w  =  D  /\  ph ) ) )
81, 6, 73bitr4i 210 . . 3  |-  ( E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ( x  =  A  /\  y  =  B  /\  E. z E. w ( z  =  C  /\  w  =  D  /\  ph )
) )
982exbii 1542 . 2  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  E. x E. y
( x  =  A  /\  y  =  B  /\  E. z E. w ( z  =  C  /\  w  =  D  /\  ph )
) )
10 ceqsex4v.1 . . 3  |-  A  e. 
_V
11 ceqsex4v.2 . . 3  |-  B  e. 
_V
12 ceqsex4v.7 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
13123anbi3d 1254 . . . 4  |-  ( x  =  A  ->  (
( z  =  C  /\  w  =  D  /\  ph )  <->  ( z  =  C  /\  w  =  D  /\  ps )
) )
14132exbidv 1796 . . 3  |-  ( x  =  A  ->  ( E. z E. w ( z  =  C  /\  w  =  D  /\  ph )  <->  E. z E. w
( z  =  C  /\  w  =  D  /\  ps ) ) )
15 ceqsex4v.8 . . . . 5  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
16153anbi3d 1254 . . . 4  |-  ( y  =  B  ->  (
( z  =  C  /\  w  =  D  /\  ps )  <->  ( z  =  C  /\  w  =  D  /\  ch )
) )
17162exbidv 1796 . . 3  |-  ( y  =  B  ->  ( E. z E. w ( z  =  C  /\  w  =  D  /\  ps )  <->  E. z E. w
( z  =  C  /\  w  =  D  /\  ch ) ) )
1810, 11, 14, 17ceqsex2v 2660 . 2  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  E. z E. w ( z  =  C  /\  w  =  D  /\  ph ) )  <->  E. z E. w ( z  =  C  /\  w  =  D  /\  ch )
)
19 ceqsex4v.3 . . 3  |-  C  e. 
_V
20 ceqsex4v.4 . . 3  |-  D  e. 
_V
21 ceqsex4v.9 . . 3  |-  ( z  =  C  ->  ( ch 
<->  th ) )
22 ceqsex4v.10 . . 3  |-  ( w  =  D  ->  ( th 
<->  ta ) )
2319, 20, 21, 22ceqsex2v 2660 . 2  |-  ( E. z E. w ( z  =  C  /\  w  =  D  /\  ch )  <->  ta )
249, 18, 233bitri 204 1  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  ceqsex8v  2664
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