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Theorem eeeanv 1851
Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
eeeanv  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
Distinct variable groups:    ph, y    ph, z    x, z, ps    x, y, ch
Allowed substitution hints:    ph( x)    ps( y)    ch( z)

Proof of Theorem eeeanv
StepHypRef Expression
1 df-3an 922 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
213exbii 1539 . 2  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x E. y E. z ( ( ph  /\  ps )  /\  ch ) )
3 eeanv 1850 . . 3  |-  ( E. y E. z ( ( ph  /\  ps )  /\  ch )  <->  ( E. y ( ph  /\  ps )  /\  E. z ch ) )
43exbii 1537 . 2  |-  ( E. x E. y E. z ( ( ph  /\ 
ps )  /\  ch ) 
<->  E. x ( E. y ( ph  /\  ps )  /\  E. z ch ) )
5 eeanv 1850 . . . 4  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
65anbi1i 446 . . 3  |-  ( ( E. x E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( ( E. x ph  /\ 
E. y ps )  /\  E. z ch )
)
7 19.41v 1825 . . 3  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x E. y
( ph  /\  ps )  /\  E. z ch )
)
8 df-3an 922 . . 3  |-  ( ( E. x ph  /\  E. y ps  /\  E. z ch )  <->  ( ( E. x ph  /\  E. y ps )  /\  E. z ch ) )
96, 7, 83bitr4i 210 . 2  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
102, 4, 93bitri 204 1  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 920   E.wex 1422
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391
This theorem is referenced by:  vtocl3  2664  spc3egv  2698  spc3gv  2699  eloprabga  5643  prarloc  6791
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