MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.42vvv Structured version   Unicode version

Theorem 19.42vvv 1804
Description: Version of 19.42 2002 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
Assertion
Ref Expression
19.42vvv  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
Distinct variable groups:    ph, x    ph, y    ph, z
Allowed substitution hints:    ps( x, y, z)

Proof of Theorem 19.42vvv
StepHypRef Expression
1 19.42vv 1803 . . 3  |-  ( E. y E. z (
ph  /\  ps )  <->  (
ph  /\  E. y E. z ps ) )
21exbii 1690 . 2  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  E. x ( ph  /\ 
E. y E. z ps ) )
3 19.42v 1801 . 2  |-  ( E. x ( ph  /\  E. y E. z ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
42, 3bitri 251 1  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369   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-ex 1636
This theorem is referenced by:  ceqsex6v  3103
  Copyright terms: Public domain W3C validator