MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3exdistr Structured version   Unicode version

Theorem 3exdistr 1806
Description: Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
3exdistr  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x
( ph  /\  E. y
( ps  /\  E. z ch ) ) )
Distinct variable groups:    ph, y    ph, z    ps, z
Allowed substitution hints:    ph( x)    ps( x, y)    ch( x, y, z)

Proof of Theorem 3exdistr
StepHypRef Expression
1 3anass 980 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  ( ps  /\  ch ) ) )
212exbii 1691 . . 3  |-  ( E. y E. z (
ph  /\  ps  /\  ch ) 
<->  E. y E. z
( ph  /\  ( ps  /\  ch ) ) )
3 19.42vv 1803 . . 3  |-  ( E. y E. z (
ph  /\  ( ps  /\ 
ch ) )  <->  ( ph  /\ 
E. y E. z
( ps  /\  ch ) ) )
4 exdistr 1802 . . . 4  |-  ( E. y E. z ( ps  /\  ch )  <->  E. y ( ps  /\  E. z ch ) )
54anbi2i 694 . . 3  |-  ( (
ph  /\  E. y E. z ( ps  /\  ch ) )  <->  ( ph  /\ 
E. y ( ps 
/\  E. z ch )
) )
62, 3, 53bitri 273 . 2  |-  ( E. y E. z (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  E. y ( ps  /\  E. z ch ) ) )
76exbii 1690 1  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ch )  <->  E. x
( ph  /\  E. y
( ps  /\  E. z ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    /\ w3a 976   E.wex 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773
This theorem depends on definitions:  df-bi 187  df-an 371  df-3an 978  df-ex 1636
This theorem is referenced by:  4exdistr  1807
  Copyright terms: Public domain W3C validator