Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm13.14 Unicode version

Theorem pm13.14 26963
Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.14  |-  ( (
[. A  /  x ]. ph  /\  -.  ph )  ->  x  =/=  A
)

Proof of Theorem pm13.14
StepHypRef Expression
1 sbceq1a 2962 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21biimprcd 218 . . 3  |-  ( [. A  /  x ]. ph  ->  ( x  =  A  ->  ph ) )
32necon3bd 2456 . 2  |-  ( [. A  /  x ]. ph  ->  ( -.  ph  ->  x  =/= 
A ) )
43imp 420 1  |-  ( (
[. A  /  x ]. ph  /\  -.  ph )  ->  x  =/=  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    =/= wne 2419   [.wsbc 2952
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-gen 1536  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-ne 2421  df-sbc 2953
  Copyright terms: Public domain W3C validator