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Theorem sgplpte21a 26133
Description: The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
sgplpte21a.2  |-  B  =  (btw `  G )
sgplpte21a.6  |-  ( ph  ->  Y  e.  P )
sgplpte21a.7  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
sgplpte21a  |-  ( ph  ->  A. z ( z  e.  ( X S Y )  <->  ( z  e.  P  /\  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
) ) )
Distinct variable groups:    z, P    z, S    z, X    z, Y    z, G    ph, z
Allowed substitution hint:    B( z)

Proof of Theorem sgplpte21a
StepHypRef Expression
1 sgplpte.1 . . 3  |-  P  =  (PPoints `  G )
2 sgplpte.3 . . 3  |-  S  =  ( seg `  G
)
3 sgplpte.4 . . 3  |-  ( ph  ->  G  e. Ibg )
4 sgplpte.5 . . 3  |-  ( ph  ->  X  e.  P )
5 sgplpte21a.2 . . 3  |-  B  =  (btw `  G )
6 sgplpte21a.6 . . 3  |-  ( ph  ->  Y  e.  P )
7 sgplpte21a.7 . . 3  |-  ( ph  ->  X  =/=  Y )
81, 2, 3, 4, 5, 6, 7sgplpte21 26132 . 2  |-  ( ph  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
9 nfcv 2419 . . . 4  |-  F/_ z
( X S Y )
10 nfrab1 2720 . . . 4  |-  F/_ z { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }
119, 10cleqf 2443 . . 3  |-  ( ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }  <->  A. z
( z  e.  ( X S Y )  <-> 
z  e.  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ) )
12 rabid 2716 . . . . . 6  |-  ( z  e.  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }  <->  ( z  e.  P  /\  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
) )
1312a1i 10 . . . . 5  |-  ( ph  ->  ( z  e.  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }  <->  ( z  e.  P  /\  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
) ) )
1413bibi2d 309 . . . 4  |-  ( ph  ->  ( ( z  e.  ( X S Y )  <->  z  e.  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )  <->  ( z  e.  ( X S Y )  <->  ( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) ) ) )
1514albidv 1611 . . 3  |-  ( ph  ->  ( A. z ( z  e.  ( X S Y )  <->  z  e.  { z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )  <->  A. z
( z  e.  ( X S Y )  <-> 
( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) ) ) )
1611, 15syl5bb 248 . 2  |-  ( ph  ->  ( ( X S Y )  =  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }  <->  A. z
( z  e.  ( X S Y )  <-> 
( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) ) ) )
178, 16mpbid 201 1  |-  ( ph  ->  A. z ( z  e.  ( X S Y )  <->  ( z  e.  P  /\  (
z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107   segcseg 26130
This theorem is referenced by:  sgplpte21b  26134
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-seg2 26131
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