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Theorem pm13.14 27008
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 3002 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21biimprcd 218 . . 3  |-  ( [. A  /  x ]. ph  ->  ( x  =  A  ->  ph ) )
32necon3bd 2484 . 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 1628    =/= wne 2447   [.wsbc 2992
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-11 1719  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1534  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-ne 2449  df-sbc 2993
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