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Theorem ee4anv 2095
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 1944 . . 3  |-  ( E. y E. z E. w ( ph  /\  ps )  <->  E. z E. y E. w ( ph  /\  ps ) )
21exbii 1726 . 2  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  E. x E. z E. y E. w (
ph  /\  ps )
)
3 eeanv 2093 . . 3  |-  ( E. y E. w (
ph  /\  ps )  <->  ( E. y ph  /\  E. w ps ) )
432exbii 1727 . 2  |-  ( E. x E. z E. y E. w (
ph  /\  ps )  <->  E. x E. z ( E. y ph  /\  E. w ps ) )
5 eeanv 2093 . 2  |-  ( E. x E. z ( E. y ph  /\  E. w ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
62, 4, 53bitri 279 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 setvar class
Syntax hints:    <-> wb 189    /\ wa 376   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-ex 1672  df-nf 1676
This theorem is referenced by:  cgsex4g  3068  5oalem7  27394  3oalem3  27398  elfuns  30753
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