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Theorem eeeanv 1919
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 936 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
213exbii 1584 . 2  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x E. y E. z ( ( ph  /\  ps )  /\  ch ) )
3 eeanv 1918 . . 3  |-  ( E. y E. z ( ( ph  /\  ps )  /\  ch )  <->  ( E. y ( ph  /\  ps )  /\  E. z ch ) )
43exbii 1582 . 2  |-  ( E. x E. y E. z ( ( ph  /\ 
ps )  /\  ch ) 
<->  E. x ( E. y ( ph  /\  ps )  /\  E. z ch ) )
5 eeanv 1918 . . . 4  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
65anbi1i 676 . . 3  |-  ( ( E. x E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( ( E. x ph  /\ 
E. y ps )  /\  E. z ch )
)
7 19.41v 1906 . . 3  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x E. y
( ph  /\  ps )  /\  E. z ch )
)
8 df-3an 936 . . 3  |-  ( ( E. x ph  /\  E. y ps  /\  E. z ch )  <->  ( ( E. x ph  /\  E. y ps )  /\  E. z ch ) )
96, 7, 83bitr4i 268 . 2  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
102, 4, 93bitri 262 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:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1541
This theorem is referenced by:  vtocl3  2916  spc3egv  2948  eloprabga  6018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545
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