MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm11.07 Unicode version

Theorem pm11.07 2075
Description: Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
Assertion
Ref Expression
pm11.07  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    ph, x, y, z    x, w, z
Allowed substitution hint:    ph( w)

Proof of Theorem pm11.07
StepHypRef Expression
1 a9e 1817 . . . . . . 7  |-  E. x  x  =  w
2 a9e 1817 . . . . . . 7  |-  E. z 
z  =  y
31, 2pm3.2i 443 . . . . . 6  |-  ( E. x  x  =  w  /\  E. z  z  =  y )
4 a9e 1817 . . . . . . 7  |-  E. x  x  =  y
5 a9e 1817 . . . . . . 7  |-  E. z 
z  =  w
64, 5pm3.2i 443 . . . . . 6  |-  ( E. x  x  =  y  /\  E. z  z  =  w )
73, 62th 232 . . . . 5  |-  ( ( E. x  x  =  w  /\  E. z 
z  =  y )  <-> 
( E. x  x  =  y  /\  E. z  z  =  w
) )
8 eeanv 2055 . . . . 5  |-  ( E. x E. z ( x  =  w  /\  z  =  y )  <->  ( E. x  x  =  w  /\  E. z 
z  =  y ) )
9 eeanv 2055 . . . . 5  |-  ( E. x E. z ( x  =  y  /\  z  =  w )  <->  ( E. x  x  =  y  /\  E. z 
z  =  w ) )
107, 8, 93bitr4i 270 . . . 4  |-  ( E. x E. z ( x  =  w  /\  z  =  y )  <->  E. x E. z ( x  =  y  /\  z  =  w )
)
1110anbi1i 679 . . 3  |-  ( ( E. x E. z
( x  =  w  /\  z  =  y )  /\  ph )  <->  ( E. x E. z
( x  =  y  /\  z  =  w )  /\  ph )
)
12 19.41vv 2035 . . 3  |-  ( E. x E. z ( ( x  =  w  /\  z  =  y )  /\  ph )  <->  ( E. x E. z
( x  =  w  /\  z  =  y )  /\  ph )
)
13 19.41vv 2035 . . 3  |-  ( E. x E. z ( ( x  =  y  /\  z  =  w )  /\  ph )  <->  ( E. x E. z
( x  =  y  /\  z  =  w )  /\  ph )
)
1411, 12, 133bitr4i 270 . 2  |-  ( E. x E. z ( ( x  =  w  /\  z  =  y )  /\  ph )  <->  E. x E. z ( ( x  =  y  /\  z  =  w )  /\  ph )
)
15 2sb5 2072 . 2  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  E. x E. z
( ( x  =  w  /\  z  =  y )  /\  ph ) )
16 2sb5 2072 . 2  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  E. x E. z
( ( x  =  y  /\  z  =  w )  /\  ph ) )
1714, 15, 163bitr4i 270 1  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619   [wsb 1882
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-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883
  Copyright terms: Public domain W3C validator