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Theorem 3dimlem1 29647
Description: Lemma for 3dim1 29656. (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 2454 . . 3  |-  ( P  =  Q  ->  ( P  =/=  R  <->  Q  =/=  R ) )
2 oveq1 5865 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  R )  =  ( Q  .\/  R
) )
32breq2d 4035 . . . 4  |-  ( P  =  Q  ->  ( S  .<_  ( P  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
43notbid 285 . . 3  |-  ( P  =  Q  ->  ( -.  S  .<_  ( P 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
52oveq1d 5873 . . . . 5  |-  ( P  =  Q  ->  (
( P  .\/  R
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
65breq2d 4035 . . . 4  |-  ( P  =  Q  ->  ( T  .<_  ( ( P 
.\/  R )  .\/  S )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
76notbid 285 . . 3  |-  ( P  =  Q  ->  ( -.  T  .<_  ( ( P  .\/  R ) 
.\/  S )  <->  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )
81, 4, 73anbi123d 1252 . 2  |-  ( P  =  Q  ->  (
( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R )  /\  -.  T  .<_  ( ( P  .\/  R ) 
.\/  S ) )  <-> 
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) ) )
98biimparc 473 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 358    /\ w3a 934    = wceq 1623    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453
This theorem is referenced by:  3dim1  29656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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