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Theorem pm13.14 27712
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 3014 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21biimprcd 216 . . 3  |-  ( [. A  /  x ]. ph  ->  ( x  =  A  ->  ph ) )
32necon3bd 2496 . 2  |-  ( [. A  /  x ]. ph  ->  ( -.  ph  ->  x  =/= 
A ) )
43imp 418 1  |-  ( (
[. A  /  x ]. ph  /\  -.  ph )  ->  x  =/=  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459   [.wsbc 3004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-sbc 3005
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