MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.41vvvv Unicode version

Theorem 19.41vvvv 2038
Description: Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
19.41vvvv  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
Distinct variable groups:    ps, w    ps, x    ps, y    ps, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 19.41vvvv
StepHypRef Expression
1 19.41vvv 2037 . . 3  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( E. x E. y E. z ph  /\ 
ps ) )
21exbii 1580 . 2  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  E. w
( E. x E. y E. z ph  /\  ps ) )
3 19.41v 2035 . 2  |-  ( E. w ( E. x E. y E. z ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
42, 3bitri 242 1  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537
This theorem is referenced by:  elo  24407
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
  Copyright terms: Public domain W3C validator