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Theorem lukshefth1 1450
Description: Lemma for renicax 1452. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
lukshefth1  |-  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) )

Proof of Theorem lukshefth1
StepHypRef Expression
1 lukshef-ax1 1449 . 2  |-  ( (
ph  -/\  ( ps  -/\  ch ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( ta 
-/\  ps )  -/\  (
( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) )
2 lukshef-ax1 1449 . . . 4  |-  ( ( ta  -/\  ( ta  -/\ 
ta ) )  -/\  ( ( th  -/\  ( th  -/\  th ) ) 
-/\  ( ( th 
-/\  ta )  -/\  (
( ta  -/\  th )  -/\  ( ta  -/\  th )
) ) ) )
3 lukshef-ax1 1449 . . . 4  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ( th  -/\  th )
)  -/\  ( ( th  -/\  ta )  -/\  ( ( ta  -/\  th )  -/\  ( ta  -/\ 
th ) ) ) ) )  -/\  (
( ( ( ta 
-/\  ps )  -/\  (
( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  (
( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  (
( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) 
-/\  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  (
( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) )  -/\  ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) ) ) )
42, 3nic-mp 1426 . . 3  |-  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  (
( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) )  -/\  ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) )
5 lukshef-ax1 1449 . . 3  |-  ( ( ( ( ( ta 
-/\  ps )  -/\  (
( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  (
( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) )  -/\  ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) )  -/\  ( (
( ph  -/\  ( ps 
-/\  ch ) )  -/\  ( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ph  -/\  ( ps  -/\  ch ) ) ) )  -/\  (
( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) 
-/\  ( ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) )  -/\  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) ) ) ) ) )
64, 5nic-mp 1426 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ch ) ) 
-/\  ( ( ta 
-/\  ( ta  -/\  ta ) )  -/\  (
( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) ) ) ) 
-/\  ( ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) )  -/\  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) ) ) )
71, 6nic-mp 1426 1  |-  ( ( ( ( ta  -/\  ps )  -/\  ( ( ph  -/\  ta )  -/\  ( ph  -/\  ta )
) )  -/\  ( th  -/\  ( th  -/\  th )
) )  -/\  ( ph  -/\  ( ps  -/\  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -/\ wnan 1287
This theorem is referenced by:  lukshefth2  1451  renicax  1452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288
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