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Theorem pm13.196a 27582
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 2200 . 2  |-  ( -. 
ph 
<->  E. y ( y  =  x  /\  [
y  /  x ]  -.  ph ) )
2 sb56 2173 . 2  |-  ( E. y ( y  =  x  /\  [ y  /  x ]  -.  ph )  <->  A. y ( y  =  x  ->  [ y  /  x ]  -.  ph ) )
3 sbn 2117 . . . . 5  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
43imbi2i 304 . . . 4  |-  ( ( y  =  x  ->  [ y  /  x ]  -.  ph )  <->  ( y  =  x  ->  -.  [
y  /  x ] ph ) )
5 con2b 325 . . . 4  |-  ( ( y  =  x  ->  -.  [ y  /  x ] ph )  <->  ( [
y  /  x ] ph  ->  -.  y  =  x ) )
6 df-ne 2600 . . . . . 6  |-  ( y  =/=  x  <->  -.  y  =  x )
76bicomi 194 . . . . 5  |-  ( -.  y  =  x  <->  y  =/=  x )
87imbi2i 304 . . . 4  |-  ( ( [ y  /  x ] ph  ->  -.  y  =  x )  <->  ( [
y  /  x ] ph  ->  y  =/=  x
) )
94, 5, 83bitri 263 . . 3  |-  ( ( y  =  x  ->  [ y  /  x ]  -.  ph )  <->  ( [
y  /  x ] ph  ->  y  =/=  x
) )
109albii 1575 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ]  -.  ph ) 
<-> 
A. y ( [ y  /  x ] ph  ->  y  =/=  x
) )
111, 2, 103bitri 263 1  |-  ( -. 
ph 
<-> 
A. y ( [ y  /  x ] ph  ->  y  =/=  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   [wsb 1658    =/= wne 2598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-ne 2600
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