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Theorem pm13.196a 36149
Description: Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.196a  |-  ( -. 
ph 
<-> 
A. y ( [ y  /  x ] ph  ->  y  =/=  x
) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem pm13.196a
StepHypRef Expression
1 sbelx 2226 . 2  |-  ( -. 
ph 
<->  E. y ( y  =  x  /\  [
y  /  x ]  -.  ph ) )
2 sb56 2196 . 2  |-  ( E. y ( y  =  x  /\  [ y  /  x ]  -.  ph )  <->  A. y ( y  =  x  ->  [ y  /  x ]  -.  ph ) )
3 sbn 2156 . . . . 5  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
43imbi2i 310 . . . 4  |-  ( ( y  =  x  ->  [ y  /  x ]  -.  ph )  <->  ( y  =  x  ->  -.  [
y  /  x ] ph ) )
5 con2b 332 . . . 4  |-  ( ( y  =  x  ->  -.  [ y  /  x ] ph )  <->  ( [
y  /  x ] ph  ->  -.  y  =  x ) )
6 df-ne 2600 . . . . . 6  |-  ( y  =/=  x  <->  -.  y  =  x )
76bicomi 202 . . . . 5  |-  ( -.  y  =  x  <->  y  =/=  x )
87imbi2i 310 . . . 4  |-  ( ( [ y  /  x ] ph  ->  -.  y  =  x )  <->  ( [
y  /  x ] ph  ->  y  =/=  x
) )
94, 5, 83bitri 271 . . 3  |-  ( ( y  =  x  ->  [ y  /  x ]  -.  ph )  <->  ( [
y  /  x ] ph  ->  y  =/=  x
) )
109albii 1661 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ]  -.  ph ) 
<-> 
A. y ( [ y  /  x ] ph  ->  y  =/=  x
) )
111, 2, 103bitri 271 1  |-  ( -. 
ph 
<-> 
A. y ( [ y  /  x ] ph  ->  y  =/=  x
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403   E.wex 1633   [wsb 1763    =/= wne 2598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764  df-ne 2600
This theorem is referenced by: (None)
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