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

Theorem ee4anv 1940
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 1756 . . 3  |-  ( E. y E. z E. w ( ph  /\  ps )  <->  E. z E. y E. w ( ph  /\  ps ) )
21exbii 1592 . 2  |-  ( E. x E. y E. z E. w (
ph  /\  ps )  <->  E. x E. z E. y E. w (
ph  /\  ps )
)
3 eeanv 1937 . . 3  |-  ( E. y E. w (
ph  /\  ps )  <->  ( E. y ph  /\  E. w ps ) )
432exbii 1593 . 2  |-  ( E. x E. z E. y E. w (
ph  /\  ps )  <->  E. x E. z ( E. y ph  /\  E. w ps ) )
5 eeanv 1937 . 2  |-  ( E. x E. z ( E. y ph  /\  E. w ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
62, 4, 53bitri 263 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 177    /\ wa 359   E.wex 1550
This theorem is referenced by:  cgsex4g  2989  th3qlem1  7010  5oalem7  23162  3oalem3  23166  elfuns  25760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator