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Theorem pm13.14 27609
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 3001 . . . 4  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
21biimprcd 216 . . 3  |-  ( [. A  /  x ]. ph  ->  ( x  =  A  ->  ph ) )
32necon3bd 2483 . 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 1623    =/= wne 2446   [.wsbc 2991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-sbc 2992
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