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Theorem nonconne 2651
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
Assertion
Ref Expression
nonconne  |-  -.  ( A  =  B  /\  A  =/=  B )

Proof of Theorem nonconne
StepHypRef Expression
1 fal 1390 . 2  |-  -. F.
2 eqneqall 2650 . . 3  |-  ( A  =  B  ->  ( A  =/=  B  -> F.  ) )
32imp 429 . 2  |-  ( ( A  =  B  /\  A  =/=  B )  -> F.  )
41, 3mto 176 1  |-  -.  ( A  =  B  /\  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1383   F. wfal 1388    =/= wne 2638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-fal 1389  df-ne 2640
This theorem is referenced by:  osumcllem11N  35565  pexmidlem8N  35576  dochexmidlem8  37069
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