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Theorem ee4anv 1856
Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
Assertion
Ref Expression
ee4anv  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
Distinct variable groups:    ph, z    ph, w    ps, x    ps, y    y, z   
x, w
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem ee4anv
StepHypRef Expression
1 excom 1786 . . 3  |-  ( E. y E. z E. w ( ph  /\  ps )  <->  E. z E. y E. w ( ph  /\  ps ) )
21exbii 1569 . 2  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  E. x E. z E. y E. w (
ph  /\  ps )
)
3 eeanv 1854 . . 3  |-  ( E. y E. w (
ph  /\  ps )  <->  ( E. y ph  /\  E. w ps ) )
432exbii 1570 . 2  |-  ( E. x E. z E. y E. w (
ph  /\  ps )  <->  E. x E. z ( E. y ph  /\  E. w ps ) )
5 eeanv 1854 . 2  |-  ( E. x E. z ( E. y ph  /\  E. w ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
62, 4, 53bitri 262 1  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528
This theorem is referenced by:  cgsex4g  2821  th3qlem1  6764  5oalem7  22239  3oalem3  22243
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-tru 1310  df-ex 1529  df-nf 1532
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