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Theorem ceqsex4v 3147
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 1782 . . . 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 975 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B )  /\  (
( z  =  C  /\  w  =  D )  /\  ph )
) )
3 df-3an 973 . . . . . . 7  |-  ( ( z  =  C  /\  w  =  D  /\  ph )  <->  ( ( z  =  C  /\  w  =  D )  /\  ph ) )
43anbi2i 692 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D  /\  ph )
)  <->  ( ( x  =  A  /\  y  =  B )  /\  (
( z  =  C  /\  w  =  D )  /\  ph )
) )
52, 4bitr4i 252 . . . . 5  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D  /\  ph ) ) )
652exbii 1673 . . . 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 973 . . . 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 277 . . 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 1673 . 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 1303 . . . 4  |-  ( x  =  A  ->  (
( z  =  C  /\  w  =  D  /\  ph )  <->  ( z  =  C  /\  w  =  D  /\  ps )
) )
14132exbidv 1721 . . 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 1303 . . . 4  |-  ( y  =  B  ->  (
( z  =  C  /\  w  =  D  /\  ps )  <->  ( z  =  C  /\  w  =  D  /\  ch )
) )
17162exbidv 1721 . . 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 3145 . 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 3145 . 2  |-  ( E. z E. w ( z  =  C  /\  w  =  D  /\  ch )  <->  ta )
249, 18, 233bitri 271 1  |-  ( E. x E. y E. z E. w ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D )  /\  ph ) 
<->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by:  ceqsex8v  3149  dihopelvalcpre  37372
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