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Theorem 3dimlem1 33410
Description: Lemma for 3dim1 33419. (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 2729 . . 3  |-  ( P  =  Q  ->  ( P  =/=  R  <->  Q  =/=  R ) )
2 oveq1 6199 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
32breq2d 4404 . . . 4  |-  ( P  =  Q  ->  ( S  .<_  ( P  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
43notbid 294 . . 3  |-  ( P  =  Q  ->  ( -.  S  .<_  ( P 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
52oveq1d 6207 . . . . 5  |-  ( P  =  Q  ->  (
( P  .\/  R
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
65breq2d 4404 . . . 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 1290 . 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 965    = wceq 1370    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   Atomscatm 33216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by:  3dim1  33419
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