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Theorem eeeanv 2094
Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
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    ps, x    ps, z    ch, x    ch, y
Allowed substitution hints:    ph( x)    ps( y)    ch( z)

Proof of Theorem eeeanv
StepHypRef Expression
1 eeanv 2093 . . 3  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
21anbi1i 709 . 2  |-  ( ( E. x E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( ( E. x ph  /\ 
E. y ps )  /\  E. z ch )
)
3 df-3an 1009 . . . . . 6  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
43exbii 1726 . . . . 5  |-  ( E. z ( ph  /\  ps  /\  ch )  <->  E. z
( ( ph  /\  ps )  /\  ch )
)
5 19.42v 1842 . . . . 5  |-  ( E. z ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  E. z ch ) )
64, 5bitri 257 . . . 4  |-  ( E. z ( ph  /\  ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  E. z ch ) )
762exbii 1727 . . 3  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x E. y ( ( ph  /\ 
ps )  /\  E. z ch ) )
8 nfv 1769 . . . . . 6  |-  F/ y ch
98nfex 2050 . . . . 5  |-  F/ y E. z ch
10919.41 2070 . . . 4  |-  ( E. y ( ( ph  /\ 
ps )  /\  E. z ch )  <->  ( E. y ( ph  /\  ps )  /\  E. z ch ) )
1110exbii 1726 . . 3  |-  ( E. x E. y ( ( ph  /\  ps )  /\  E. z ch )  <->  E. x ( E. y ( ph  /\  ps )  /\  E. z ch ) )
12 nfv 1769 . . . . 5  |-  F/ x ch
1312nfex 2050 . . . 4  |-  F/ x E. z ch
141319.41 2070 . . 3  |-  ( E. x ( E. y
( ph  /\  ps )  /\  E. z ch )  <->  ( E. x E. y
( ph  /\  ps )  /\  E. z ch )
)
157, 11, 143bitri 279 . 2  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  ( E. x E. y ( ph  /\ 
ps )  /\  E. z ch ) )
16 df-3an 1009 . 2  |-  ( ( E. x ph  /\  E. y ps  /\  E. z ch )  <->  ( ( E. x ph  /\  E. y ps )  /\  E. z ch ) )
172, 15, 163bitr4i 285 1  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-ex 1672  df-nf 1676
This theorem is referenced by:  vtocl3  3089  spc3egv  3124  eloprabga  6402
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