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Theorem isibg2a1 26119
Description: Two distinct points  X,  Y have a point before them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
isibg2a.1  |-  P  =  (PPoints `  G )
isibg2a.2  |-  B  =  (btw `  G )
isibg2a.3  |-  ( ph  ->  G  e. Ibg )
isibg2a.4  |-  ( ph  ->  X  e.  P )
isibg2a.5  |-  ( ph  ->  Y  e.  P )
isibg2a1.6  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
isibg2a1  |-  ( ph  ->  E. a  e.  P  X  e.  ( a B Y ) )
Distinct variable groups:    B, a    P, a    X, a    Y, a
Allowed substitution hints:    ph( a)    G( a)

Proof of Theorem isibg2a1
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg2a.1 . . 3  |-  P  =  (PPoints `  G )
2 isibg2a.2 . . 3  |-  B  =  (btw `  G )
3 isibg2a.3 . . 3  |-  ( ph  ->  G  e. Ibg )
4 isibg2a.4 . . 3  |-  ( ph  ->  X  e.  P )
5 isibg2a.5 . . 3  |-  ( ph  ->  Y  e.  P )
6 isibg2a1.6 . . 3  |-  ( ph  ->  X  =/=  Y )
71, 2, 3, 4, 5, 6isibg2a 26118 . 2  |-  ( ph  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  (
a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )
8 simp1 955 . . . . 5  |-  ( ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) )  ->  X  e.  ( a B Y ) )
98rexlimivw 2663 . . . 4  |-  ( E. c  e.  P  ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) )  ->  X  e.  ( a B Y ) )
109rexlimivw 2663 . . 3  |-  ( E. b  e.  P  E. c  e.  P  ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) )  ->  X  e.  ( a B Y ) )
1110reximi 2650 . 2  |-  ( E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) )  ->  E. a  e.  P  X  e.  ( a B Y ) )
127, 11syl 15 1  |-  ( ph  ->  E. a  e.  P  X  e.  ( a B Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107
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  ax-nul 4149
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-ibg2 26109
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