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Theorem ee4anv 1868
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 1798 . . 3  |-  ( E. y E. z E. w ( ph  /\  ps )  <->  E. z E. y E. w ( ph  /\  ps ) )
21exbii 1572 . 2  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  E. x E. z E. y E. w (
ph  /\  ps )
)
3 eeanv 1866 . . 3  |-  ( E. y E. w (
ph  /\  ps )  <->  ( E. y ph  /\  E. w ps ) )
432exbii 1573 . 2  |-  ( E. x E. z E. y E. w (
ph  /\  ps )  <->  E. x E. z ( E. y ph  /\  E. w ps ) )
5 eeanv 1866 . 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 1531
This theorem is referenced by:  cgsex4g  2834  th3qlem1  6780  5oalem7  22255  3oalem3  22259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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