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Theorem dalemccnedd 29876
Description: Lemma for dath 29925. Frequently-used utility lemma. (Contributed by NM, 15-Aug-2012.)
Hypothesis
Ref Expression
da.ps0  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
Assertion
Ref Expression
dalemccnedd  |-  ( ps 
->  c  =/=  d
)

Proof of Theorem dalemccnedd
StepHypRef Expression
1 da.ps0 . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
2 simp31 991 . . 3  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  -> 
d  =/=  c )
31, 2sylbi 187 . 2  |-  ( ps 
->  d  =/=  c
)
43necomd 2529 1  |-  ( ps 
->  c  =/=  d
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858
This theorem is referenced by:  dalemswapyzps  29879  dalemrotps  29880  dalemcjden  29881
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-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-cleq 2276  df-ne 2448
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