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Theorem 3dimlem1 30256
Description: Lemma for 3dim1 30265. (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 2610 . . 3  |-  ( P  =  Q  ->  ( P  =/=  R  <->  Q  =/=  R ) )
2 oveq1 6089 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
32breq2d 4225 . . . 4  |-  ( P  =  Q  ->  ( S  .<_  ( P  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
43notbid 287 . . 3  |-  ( P  =  Q  ->  ( -.  S  .<_  ( P 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
52oveq1d 6097 . . . . 5  |-  ( P  =  Q  ->  (
( P  .\/  R
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
65breq2d 4225 . . . 4  |-  ( P  =  Q  ->  ( T  .<_  ( ( P 
.\/  R )  .\/  S )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
76notbid 287 . . 3  |-  ( P  =  Q  ->  ( -.  T  .<_  ( ( P  .\/  R ) 
.\/  S )  <->  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )
81, 4, 73anbi123d 1255 . 2  |-  ( P  =  Q  ->  (
( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R )  /\  -.  T  .<_  ( ( P  .\/  R ) 
.\/  S ) )  <-> 
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) ) )
98biimparc 475 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Atomscatm 30062
This theorem is referenced by:  3dim1  30265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085
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