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Theorem el2spthsoton 25006
 Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
el2spthsoton 2SPathOnOt 2SPathOnOt
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem el2spthsoton
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 2spthsot 24995 . . 3 2SPathOnOt 2SPathOnOt
21eleq2d 2527 . 2 2SPathOnOt 2SPathOnOt
3 eleq1 2529 . . . . 5 2SPathOnOt 2SPathOnOt
432rexbidv 2975 . . . 4 2SPathOnOt 2SPathOnOt
54elrab 3257 . . 3 2SPathOnOt 2SPathOnOt
65a1i 11 . 2 2SPathOnOt 2SPathOnOt
7 simpr 461 . . 3 2SPathOnOt 2SPathOnOt
8 simpr 461 . . . . 5 2SPathOnOt 2SPathOnOt
9 2spthonot3v 25003 . . . . . . . 8 2SPathOnOt
109simp3d 1010 . . . . . . 7 2SPathOnOt
1110a1i 11 . . . . . 6 2SPathOnOt
1211rexlimivv 2954 . . . . 5 2SPathOnOt
138, 12jccil 540 . . . 4 2SPathOnOt 2SPathOnOt
1413ex 434 . . 3 2SPathOnOt 2SPathOnOt
157, 14impbid2 204 . 2 2SPathOnOt 2SPathOnOt
162, 6, 153bitrd 279 1 2SPathOnOt 2SPathOnOt
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wrex 2808  crab 2811  cvv 3109   cxp 5006  (class class class)co 6296   2SPathOnOt c2spthot 24983   2SPathOnOt c2pthonot 24984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-2spthonot 24987  df-2spthsot 24988 This theorem is referenced by:  el2pthsot  25008  2spot2iun2spont  25018  usg2spot2nb  25192
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