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Theorem ceqsex3v 2723
Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
ceqsex3v.1  |-  A  e. 
_V
ceqsex3v.2  |-  B  e. 
_V
ceqsex3v.3  |-  C  e. 
_V
ceqsex3v.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex3v.5  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
ceqsex3v.6  |-  ( z  =  C  ->  ( ch 
<->  th ) )
Assertion
Ref Expression
ceqsex3v  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph ) 
<->  th )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ps, x    ch, y    th, z
Allowed substitution hints:    ph( x, y, z)    ps( y, z)    ch( x, z)    th( x, y)

Proof of Theorem ceqsex3v
StepHypRef Expression
1 anass 398 . . . . . 6  |-  ( ( ( x  =  A  /\  ( y  =  B  /\  z  =  C ) )  /\  ph )  <->  ( x  =  A  /\  ( ( y  =  B  /\  z  =  C )  /\  ph ) ) )
2 3anass 966 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  <->  ( x  =  A  /\  ( y  =  B  /\  z  =  C ) ) )
32anbi1i 453 . . . . . 6  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( ( x  =  A  /\  ( y  =  B  /\  z  =  C ) )  /\  ph ) )
4 df-3an 964 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  C  /\  ph )  <->  ( ( y  =  B  /\  z  =  C )  /\  ph ) )
54anbi2i 452 . . . . . 6  |-  ( ( x  =  A  /\  ( y  =  B  /\  z  =  C  /\  ph ) )  <-> 
( x  =  A  /\  ( ( y  =  B  /\  z  =  C )  /\  ph ) ) )
61, 3, 53bitr4i 211 . . . . 5  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( x  =  A  /\  ( y  =  B  /\  z  =  C  /\  ph ) ) )
762exbii 1585 . . . 4  |-  ( E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  E. y E. z ( x  =  A  /\  ( y  =  B  /\  z  =  C  /\  ph ) ) )
8 19.42vv 1883 . . . 4  |-  ( E. y E. z ( x  =  A  /\  ( y  =  B  /\  z  =  C  /\  ph ) )  <-> 
( x  =  A  /\  E. y E. z ( y  =  B  /\  z  =  C  /\  ph )
) )
97, 8bitri 183 . . 3  |-  ( E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( x  =  A  /\  E. y E. z ( y  =  B  /\  z  =  C  /\  ph ) ) )
109exbii 1584 . 2  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph ) 
<->  E. x ( x  =  A  /\  E. y E. z ( y  =  B  /\  z  =  C  /\  ph )
) )
11 ceqsex3v.1 . . . 4  |-  A  e. 
_V
12 ceqsex3v.4 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
13123anbi3d 1296 . . . . 5  |-  ( x  =  A  ->  (
( y  =  B  /\  z  =  C  /\  ph )  <->  ( y  =  B  /\  z  =  C  /\  ps )
) )
14132exbidv 1840 . . . 4  |-  ( x  =  A  ->  ( E. y E. z ( y  =  B  /\  z  =  C  /\  ph )  <->  E. y E. z
( y  =  B  /\  z  =  C  /\  ps ) ) )
1511, 14ceqsexv 2720 . . 3  |-  ( E. x ( x  =  A  /\  E. y E. z ( y  =  B  /\  z  =  C  /\  ph )
)  <->  E. y E. z
( y  =  B  /\  z  =  C  /\  ps ) )
16 ceqsex3v.2 . . . 4  |-  B  e. 
_V
17 ceqsex3v.3 . . . 4  |-  C  e. 
_V
18 ceqsex3v.5 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
19 ceqsex3v.6 . . . 4  |-  ( z  =  C  ->  ( ch 
<->  th ) )
2016, 17, 18, 19ceqsex2v 2722 . . 3  |-  ( E. y E. z ( y  =  B  /\  z  =  C  /\  ps )  <->  th )
2115, 20bitri 183 . 2  |-  ( E. x ( x  =  A  /\  E. y E. z ( y  =  B  /\  z  =  C  /\  ph )
)  <->  th )
2210, 21bitri 183 1  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph ) 
<->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by:  ceqsex6v  2725
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