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Theorem pm11.6 26990
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 1787 . . 3  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ps )  /\  ch ) )
2 an32 775 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ch )  <->  ( ( ph  /\  ch )  /\  ps ) )
322exbii 1571 . . 3  |-  ( E. y E. x ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ch )  /\  ps ) )
41, 3bitri 242 . 2  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. y E. x ( ( ph  /\ 
ch )  /\  ps ) )
5 19.41v 1843 . . 3  |-  ( E. y ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( E. y (
ph  /\  ps )  /\  ch ) )
65exbii 1570 . 2  |-  ( E. x E. y ( ( ph  /\  ps )  /\  ch )  <->  E. x
( E. y (
ph  /\  ps )  /\  ch ) )
7 19.41v 1843 . . 3  |-  ( E. x ( ( ph  /\ 
ch )  /\  ps ) 
<->  ( E. x (
ph  /\  ch )  /\  ps ) )
87exbii 1570 . 2  |-  ( E. y E. x ( ( ph  /\  ch )  /\  ps )  <->  E. y
( E. x (
ph  /\  ch )  /\  ps ) )
94, 6, 83bitr3i 268 1  |-  ( E. x ( E. y
( ph  /\  ps )  /\  ch )  <->  E. y
( E. x (
ph  /\  ch )  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533
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