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Theorem isibg2a2 26223
 Description: Two distinct points , have a point between them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
isibg2a.1 PPoints
isibg2a.2 btw
isibg2a.3 Ibg
isibg2a.4
isibg2a.5
isibg2a2.6
Assertion
Ref Expression
isibg2a2
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem isibg2a2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg2a.1 . . 3 PPoints
2 isibg2a.2 . . 3 btw
3 isibg2a.3 . . 3 Ibg
4 isibg2a.4 . . 3
5 isibg2a.5 . . 3
6 isibg2a2.6 . . 3
71, 2, 3, 4, 5, 6isibg2a 26221 . 2
8 simp2 956 . . . . 5
98rexlimivw 2676 . . . 4
109reximi 2663 . . 3
1110rexlimivw 2676 . 2
127, 11syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 934   wceq 1632   wcel 1696   wne 2459  wrex 2557  cfv 5271  (class class class)co 5874  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210 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  ax-nul 4165 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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-ibg2 26212
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