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Theorem isibg2aalem3 26218
Description: Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
Assertion
Ref Expression
isibg2aalem3  |-  ( M  e.  L  ->  ( A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) )  ->  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M )  ->  (
( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
M )  =  (/) )  ->  ( ( X B Z )  i^i 
M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i  M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) )
Distinct variable groups:    B, l    L, l    M, l    X, l    Y, l    Z, l

Proof of Theorem isibg2aalem3
StepHypRef Expression
1 neleq2 2551 . . . 4  |-  ( l  =  M  ->  ( X  e/  l  <->  X  e/  M ) )
2 neleq2 2551 . . . 4  |-  ( l  =  M  ->  ( Y  e/  l  <->  Y  e/  M ) )
3 neleq2 2551 . . . 4  |-  ( l  =  M  ->  ( Z  e/  l  <->  Z  e/  M ) )
41, 2, 33anbi123d 1252 . . 3  |-  ( l  =  M  ->  (
( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  <->  ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M ) ) )
5 ineq2 3377 . . . . . . 7  |-  ( l  =  M  ->  (
( X B Y )  i^i  l )  =  ( ( X B Y )  i^i 
M ) )
65eqeq1d 2304 . . . . . 6  |-  ( l  =  M  ->  (
( ( X B Y )  i^i  l
)  =  (/)  <->  ( ( X B Y )  i^i 
M )  =  (/) ) )
7 ineq2 3377 . . . . . . 7  |-  ( l  =  M  ->  (
( Y B Z )  i^i  l )  =  ( ( Y B Z )  i^i 
M ) )
87eqeq1d 2304 . . . . . 6  |-  ( l  =  M  ->  (
( ( Y B Z )  i^i  l
)  =  (/)  <->  ( ( Y B Z )  i^i 
M )  =  (/) ) )
96, 8anbi12d 691 . . . . 5  |-  ( l  =  M  ->  (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  <->  ( (
( X B Y )  i^i  M )  =  (/)  /\  (
( Y B Z )  i^i  M )  =  (/) ) ) )
10 ineq2 3377 . . . . . 6  |-  ( l  =  M  ->  (
( X B Z )  i^i  l )  =  ( ( X B Z )  i^i 
M ) )
1110eqeq1d 2304 . . . . 5  |-  ( l  =  M  ->  (
( ( X B Z )  i^i  l
)  =  (/)  <->  ( ( X B Z )  i^i 
M )  =  (/) ) )
129, 11imbi12d 311 . . . 4  |-  ( l  =  M  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) 
<->  ( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
M )  =  (/) )  ->  ( ( X B Z )  i^i 
M )  =  (/) ) ) )
135neeq1d 2472 . . . . . 6  |-  ( l  =  M  ->  (
( ( X B Y )  i^i  l
)  =/=  (/)  <->  ( ( X B Y )  i^i 
M )  =/=  (/) ) )
147neeq1d 2472 . . . . . 6  |-  ( l  =  M  ->  (
( ( Y B Z )  i^i  l
)  =/=  (/)  <->  ( ( Y B Z )  i^i 
M )  =/=  (/) ) )
1513, 14anbi12d 691 . . . . 5  |-  ( l  =  M  ->  (
( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
( Y B Z )  i^i  l )  =/=  (/) )  <->  ( (
( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i  M )  =/=  (/) ) ) )
1615, 11imbi12d 311 . . . 4  |-  ( l  =  M  ->  (
( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  <->  ( (
( ( X B Y )  i^i  M
)  =/=  (/)  /\  (
( Y B Z )  i^i  M )  =/=  (/) )  ->  (
( X B Z )  i^i  M )  =  (/) ) ) )
1712, 16anbi12d 691 . . 3  |-  ( l  =  M  ->  (
( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  (
( Y B Z )  i^i  l )  =  (/) )  ->  (
( X B Z )  i^i  l )  =  (/) )  /\  (
( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
( Y B Z )  i^i  l )  =/=  (/) )  ->  (
( X B Z )  i^i  l )  =  (/) ) )  <->  ( (
( ( ( X B Y )  i^i 
M )  =  (/)  /\  ( ( Y B Z )  i^i  M
)  =  (/) )  -> 
( ( X B Z )  i^i  M
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i 
M )  =/=  (/) )  -> 
( ( X B Z )  i^i  M
)  =  (/) ) ) ) )
184, 17imbi12d 311 . 2  |-  ( l  =  M  ->  (
( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) )  <->  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M )  ->  (
( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
M )  =  (/) )  ->  ( ( X B Z )  i^i 
M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i  M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) )
1918rspcv 2893 1  |-  ( M  e.  L  ->  ( A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) )  ->  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M )  ->  (
( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
M )  =  (/) )  ->  ( ( X B Z )  i^i 
M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i  M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556    i^i cin 3164   (/)c0 3468  (class class class)co 5874
This theorem is referenced by:  bsstr  26231  nbssntr  26232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-v 2803  df-in 3172
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