MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ee4anv Structured version   Unicode version

Theorem ee4anv 2049
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 1903 . . 3  |-  ( E. y E. z E. w ( ph  /\  ps )  <->  E. z E. y E. w ( ph  /\  ps ) )
21exbii 1712 . 2  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  E. x E. z E. y E. w (
ph  /\  ps )
)
3 eeanv 2047 . . 3  |-  ( E. y E. w (
ph  /\  ps )  <->  ( E. y ph  /\  E. w ps ) )
432exbii 1713 . 2  |-  ( E. x E. z E. y E. w (
ph  /\  ps )  <->  E. x E. z ( E. y ph  /\  E. w ps ) )
5 eeanv 2047 . 2  |-  ( E. x E. z ( E. y ph  /\  E. w ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
62, 4, 53bitri 274 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 187    /\ wa 370   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  cgsex4g  3116  5oalem7  27311  3oalem3  27315  elfuns  30687
  Copyright terms: Public domain W3C validator