Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.6 Unicode version

Theorem pm11.6 27591
Description: Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
pm11.6  |-  ( E. x ( E. y
( ph  /\  ps )  /\  ch )  <->  E. y
( E. x (
ph  /\  ch )  /\  ps ) )
Distinct variable groups:    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)

Proof of Theorem pm11.6
StepHypRef Expression
1 excom 1786 . . 3  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ps )  /\  ch ) )
2 an32 773 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  ( ( ph  /\  ch )  /\  ps ) )
322exbii 1570 . . 3  |-  ( E. y E. x ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ch )  /\  ps ) )
41, 3bitri 240 . 2  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ch )  /\  ps ) )
5 19.41v 1842 . . 3  |-  ( E. y ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( E. y (
ph  /\  ps )  /\  ch ) )
65exbii 1569 . 2  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. x
( E. y (
ph  /\  ps )  /\  ch ) )
7 19.41v 1842 . . 3  |-  ( E. x ( ( ph  /\ 
ch )  /\  ps ) 
<->  ( E. x (
ph  /\  ch )  /\  ps ) )
87exbii 1569 . 2  |-  ( E. y E. x ( ( ph  /\  ch )  /\  ps )  <->  E. y
( E. x (
ph  /\  ch )  /\  ps ) )
94, 6, 83bitr3i 266 1  |-  ( E. x ( E. y
( ph  /\  ps )  /\  ch )  <->  E. y
( E. x (
ph  /\  ch )  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
  Copyright terms: Public domain W3C validator