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Theorem pm11.07 2054
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 a9ev 1637 . . . . . . 7  |-  E. x  x  =  w
2 a9ev 1637 . . . . . . 7  |-  E. z 
z  =  y
31, 2pm3.2i 441 . . . . . 6  |-  ( E. x  x  =  w  /\  E. z  z  =  y )
4 a9ev 1637 . . . . . . 7  |-  E. x  x  =  y
5 a9ev 1637 . . . . . . 7  |-  E. z 
z  =  w
64, 5pm3.2i 441 . . . . . 6  |-  ( E. x  x  =  y  /\  E. z  z  =  w )
73, 62th 230 . . . . 5  |-  ( ( E. x  x  =  w  /\  E. z 
z  =  y )  <-> 
( E. x  x  =  y  /\  E. z  z  =  w
) )
8 eeanv 1854 . . . . 5  |-  ( E. x E. z ( x  =  w  /\  z  =  y )  <->  ( E. x  x  =  w  /\  E. z 
z  =  y ) )
9 eeanv 1854 . . . . 5  |-  ( E. x E. z ( x  =  y  /\  z  =  w )  <->  ( E. x  x  =  y  /\  E. z 
z  =  w ) )
107, 8, 93bitr4i 268 . . . 4  |-  ( E. x E. z ( x  =  w  /\  z  =  y )  <->  E. x E. z ( x  =  y  /\  z  =  w )
)
1110anbi1i 676 . . 3  |-  ( ( E. x E. z
( x  =  w  /\  z  =  y )  /\  ph )  <->  ( E. x E. z
( x  =  y  /\  z  =  w )  /\  ph )
)
12 19.41vv 1843 . . 3  |-  ( E. x E. z ( ( x  =  w  /\  z  =  y )  /\  ph )  <->  ( E. x E. z
( x  =  w  /\  z  =  y )  /\  ph )
)
13 19.41vv 1843 . . 3  |-  ( E. x E. z ( ( x  =  y  /\  z  =  w )  /\  ph )  <->  ( E. x E. z
( x  =  y  /\  z  =  w )  /\  ph )
)
1411, 12, 133bitr4i 268 . 2  |-  ( E. x E. z ( ( x  =  w  /\  z  =  y )  /\  ph )  <->  E. x E. z ( ( x  =  y  /\  z  =  w )  /\  ph )
)
15 2sb5 2051 . 2  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  E. x E. z
( ( x  =  w  /\  z  =  y )  /\  ph ) )
16 2sb5 2051 . 2  |-  ( [ y  /  x ] [ w  /  z ] ph  <->  E. x E. z
( ( x  =  y  /\  z  =  w )  /\  ph ) )
1714, 15, 163bitr4i 268 1  |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   [wsb 1629
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  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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