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Theorem eeeanv 1885
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 949 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
213exbii 1571 . 2  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x E. y E. z ( ( ph  /\  ps )  /\  ch ) )
3 eeanv 1884 . . 3  |-  ( E. y E. z ( ( ph  /\  ps )  /\  ch )  <->  ( E. y ( ph  /\  ps )  /\  E. z ch ) )
43exbii 1569 . 2  |-  ( E. x E. y E. z ( ( ph  /\ 
ps )  /\  ch ) 
<->  E. x ( E. y ( ph  /\  ps )  /\  E. z ch ) )
5 eeanv 1884 . . . 4  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
65anbi1i 453 . . 3  |-  ( ( E. x E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( ( E. x ph  /\ 
E. y ps )  /\  E. z ch )
)
7 19.41v 1858 . . 3  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x E. y
( ph  /\  ps )  /\  E. z ch )
)
8 df-3an 949 . . 3  |-  ( ( E. x ph  /\  E. y ps  /\  E. z ch )  <->  ( ( E. x ph  /\  E. y ps )  /\  E. z ch ) )
96, 7, 83bitr4i 211 . 2  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
102, 4, 93bitri 205 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 103    <-> wb 104    /\ w3a 947   E.wex 1453
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422
This theorem is referenced by:  vtocl3  2716  spc3egv  2751  spc3gv  2752  eloprabga  5826  prarloc  7279
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