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Theorem eeeeanv 24944
Description: Rearrange existential quantifiers. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
eeeeanv  |-  ( E. w E. x E. y E. z ( (
ph  /\  ps  /\  ch )  /\  th )  <->  ( ( E. w ph  /\  E. x ps  /\  E. y ch )  /\  E. z th ) )
Distinct variable groups:    ch, w, x, z    ph, x, y    ph, z    ps, w, y    ps, z    th, w, x, y
Allowed substitution hints:    ph( w)    ps( x)    ch( y)    th( z)

Proof of Theorem eeeeanv
StepHypRef Expression
1 19.41vvv 1844 . . . . 5  |-  ( E. x E. y E. z ( ( ps 
/\  ch  /\  th )  /\  ph )  <->  ( E. x E. y E. z
( ps  /\  ch  /\ 
th )  /\  ph ) )
2 eeeanv 1855 . . . . . 6  |-  ( E. x E. y E. z ( ps  /\  ch  /\  th )  <->  ( E. x ps  /\  E. y ch  /\  E. z th ) )
32anbi2ci 677 . . . . 5  |-  ( ( E. x E. y E. z ( ps  /\  ch  /\  th )  /\  ph )  <->  ( ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) ) )
41, 3bitri 240 . . . 4  |-  ( E. x E. y E. z ( ( ps 
/\  ch  /\  th )  /\  ph )  <->  ( ph  /\  ( E. x ps 
/\  E. y ch  /\  E. z th ) ) )
54exbii 1569 . . 3  |-  ( E. w E. x E. y E. z ( ( ps  /\  ch  /\  th )  /\  ph )  <->  E. w ( ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) ) )
6 19.41v 1842 . . 3  |-  ( E. w ( ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) )  <->  ( E. w ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) ) )
75, 6bitri 240 . 2  |-  ( E. w E. x E. y E. z ( ( ps  /\  ch  /\  th )  /\  ph )  <->  ( E. w ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) ) )
8 ancom 437 . . . . 5  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ph ) 
<->  ( ph  /\  ( ps  /\  ch  /\  th ) ) )
9 and4com 24940 . . . . 5  |-  ( (
ph  /\  ( ps  /\ 
ch  /\  th )
)  <->  ( ( ph  /\ 
ps  /\  ch )  /\  th ) )
108, 9bitri 240 . . . 4  |-  ( ( ( ps  /\  ch  /\ 
th )  /\  ph ) 
<->  ( ( ph  /\  ps  /\  ch )  /\  th ) )
11103exbii 1571 . . 3  |-  ( E. x E. y E. z ( ( ps 
/\  ch  /\  th )  /\  ph )  <->  E. x E. y E. z ( ( ph  /\  ps  /\ 
ch )  /\  th ) )
1211exbii 1569 . 2  |-  ( E. w E. x E. y E. z ( ( ps  /\  ch  /\  th )  /\  ph )  <->  E. w E. x E. y E. z ( (
ph  /\  ps  /\  ch )  /\  th ) )
13 and4com 24940 . 2  |-  ( ( E. w ph  /\  ( E. x ps  /\  E. y ch  /\  E. z th ) )  <->  ( ( E. w ph  /\  E. x ps  /\  E. y ch )  /\  E. z th ) )
147, 12, 133bitr3i 266 1  |-  ( E. w E. x E. y E. z ( (
ph  /\  ps  /\  ch )  /\  th )  <->  ( ( E. w ph  /\  E. x ps  /\  E. y ch )  /\  E. z th ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528
This theorem is referenced by:  elo  25041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532
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