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Theorem pm13.14 27524
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 3163 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21biimprcd 217 . . 3  |-  ( [. A  /  x ]. ph  ->  ( x  =  A  ->  ph ) )
32necon3bd 2635 . 2  |-  ( [. A  /  x ]. ph  ->  ( -.  ph  ->  x  =/= 
A ) )
43imp 419 1  |-  ( (
[. A  /  x ]. ph  /\  -.  ph )  ->  x  =/=  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2598   [.wsbc 3153
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-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ne 2600  df-sbc 3154
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