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Theorem 3dimlem1 34884
Description: Lemma for 3dim1 34893. (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 2722 . . 3  |-  ( P  =  Q  ->  ( P  =/=  R  <->  Q  =/=  R ) )
2 oveq1 6284 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
32breq2d 4445 . . . 4  |-  ( P  =  Q  ->  ( S  .<_  ( P  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
43notbid 294 . . 3  |-  ( P  =  Q  ->  ( -.  S  .<_  ( P 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
52oveq1d 6292 . . . . 5  |-  ( P  =  Q  ->  (
( P  .\/  R
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
65breq2d 4445 . . . 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 1298 . 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 972    = wceq 1381    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   lecple 14576   joincjn 15442   Atomscatm 34690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280
This theorem is referenced by:  3dim1  34893
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