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Theorem 3dimlem1 34129
Description: Lemma for 3dim1 34138. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dimlem1  |-  ( ( ( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  P  =  Q )  ->  ( P  =/=  R  /\  -.  S  .<_  ( P  .\/  R
)  /\  -.  T  .<_  ( ( P  .\/  R )  .\/  S ) ) )

Proof of Theorem 3dimlem1
StepHypRef Expression
1 neeq1 2741 . . 3  |-  ( P  =  Q  ->  ( P  =/=  R  <->  Q  =/=  R ) )
2 oveq1 6282 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
32breq2d 4452 . . . 4  |-  ( P  =  Q  ->  ( S  .<_  ( P  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
43notbid 294 . . 3  |-  ( P  =  Q  ->  ( -.  S  .<_  ( P 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
52oveq1d 6290 . . . . 5  |-  ( P  =  Q  ->  (
( P  .\/  R
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
65breq2d 4452 . . . 4  |-  ( P  =  Q  ->  ( T  .<_  ( ( P 
.\/  R )  .\/  S )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
76notbid 294 . . 3  |-  ( P  =  Q  ->  ( -.  T  .<_  ( ( P  .\/  R ) 
.\/  S )  <->  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )
81, 4, 73anbi123d 1294 . 2  |-  ( P  =  Q  ->  (
( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R )  /\  -.  T  .<_  ( ( P  .\/  R ) 
.\/  S ) )  <-> 
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) ) )
98biimparc 487 1  |-  ( ( ( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  P  =  Q )  ->  ( P  =/=  R  /\  -.  S  .<_  ( P  .\/  R
)  /\  -.  T  .<_  ( ( P  .\/  R )  .\/  S ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   lecple 14551   joincjn 15420   Atomscatm 33935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278
This theorem is referenced by:  3dim1  34138
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