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Theorem nb3graprlem2 25259
 Description: Lemma 2 for nb3grapr 25260. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
Assertion
Ref Expression
nb3graprlem2 USGrph Neighbors Neighbors
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem nb3graprlem2
StepHypRef Expression
1 sneq 3969 . . . . . 6
21difeq2d 3540 . . . . 5
3 preq1 4042 . . . . . 6
43eqeq2d 2481 . . . . 5 Neighbors Neighbors
52, 4rexeqbidv 2988 . . . 4 Neighbors Neighbors
6 sneq 3969 . . . . . 6
76difeq2d 3540 . . . . 5
8 preq1 4042 . . . . . 6
98eqeq2d 2481 . . . . 5 Neighbors Neighbors
107, 9rexeqbidv 2988 . . . 4 Neighbors Neighbors
11 sneq 3969 . . . . . 6
1211difeq2d 3540 . . . . 5
13 preq1 4042 . . . . . 6
1413eqeq2d 2481 . . . . 5 Neighbors Neighbors
1512, 14rexeqbidv 2988 . . . 4 Neighbors Neighbors
165, 10, 15rextpg 4015 . . 3 Neighbors Neighbors Neighbors Neighbors
17163ad2ant1 1051 . 2 USGrph Neighbors Neighbors Neighbors Neighbors
18 simpl 464 . . . 4 USGrph
19 difeq1 3533 . . . . . 6
2019adantr 472 . . . . 5 USGrph
2120rexeqdv 2980 . . . 4 USGrph Neighbors Neighbors
2218, 21rexeqbidv 2988 . . 3 USGrph Neighbors Neighbors
23223ad2ant2 1052 . 2 USGrph Neighbors Neighbors
24 preq2 4043 . . . . . . . 8
2524eqeq2d 2481 . . . . . . 7 Neighbors Neighbors
26 preq2 4043 . . . . . . . 8
2726eqeq2d 2481 . . . . . . 7 Neighbors Neighbors
2825, 27rexprg 4013 . . . . . 6 Neighbors Neighbors Neighbors
29283adant1 1048 . . . . 5 Neighbors Neighbors Neighbors
30 preq2 4043 . . . . . . . . 9
3130eqeq2d 2481 . . . . . . . 8 Neighbors Neighbors
32 preq2 4043 . . . . . . . . 9
3332eqeq2d 2481 . . . . . . . 8 Neighbors Neighbors
3431, 33rexprg 4013 . . . . . . 7 Neighbors Neighbors Neighbors
3534ancoms 460 . . . . . 6 Neighbors Neighbors Neighbors
36353adant2 1049 . . . . 5 Neighbors Neighbors Neighbors
37 preq2 4043 . . . . . . . 8
3837eqeq2d 2481 . . . . . . 7 Neighbors Neighbors
39 preq2 4043 . . . . . . . 8
4039eqeq2d 2481 . . . . . . 7 Neighbors Neighbors
4138, 40rexprg 4013 . . . . . 6 Neighbors Neighbors Neighbors
42413adant3 1050 . . . . 5 Neighbors Neighbors Neighbors
4329, 36, 423orbi123d 1364 . . . 4 Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
44433ad2ant1 1051 . . 3 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
45 tprot 4058 . . . . . . . . 9
4645a1i 11 . . . . . . . 8
4746difeq1d 3539 . . . . . . 7
48 necom 2696 . . . . . . . . 9
49 necom 2696 . . . . . . . . 9
50 diftpsn3 4101 . . . . . . . . 9
5148, 49, 50syl2anb 487 . . . . . . . 8
52513adant3 1050 . . . . . . 7
5347, 52eqtrd 2505 . . . . . 6
5453rexeqdv 2980 . . . . 5 Neighbors Neighbors
55 tprot 4058 . . . . . . . . . 10
5655eqcomi 2480 . . . . . . . . 9
5756a1i 11 . . . . . . . 8
5857difeq1d 3539 . . . . . . 7
59 necom 2696 . . . . . . . . . . . 12
6059anbi1i 709 . . . . . . . . . . 11
6160biimpi 199 . . . . . . . . . 10
6261ancoms 460 . . . . . . . . 9
63 diftpsn3 4101 . . . . . . . . 9
6462, 63syl 17 . . . . . . . 8
65643adant2 1049 . . . . . . 7
6658, 65eqtrd 2505 . . . . . 6
6766rexeqdv 2980 . . . . 5 Neighbors Neighbors
68 diftpsn3 4101 . . . . . . 7
69683adant1 1048 . . . . . 6
7069rexeqdv 2980 . . . . 5 Neighbors Neighbors
7154, 67, 703orbi123d 1364 . . . 4 Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
72713ad2ant3 1053 . . 3 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
73 prcom 4041 . . . . . . . 8
7473eqeq2i 2483 . . . . . . 7 Neighbors Neighbors
7574orbi2i 528 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
76 oridm 523 . . . . . 6 Neighbors Neighbors Neighbors
7775, 76bitr2i 258 . . . . 5 Neighbors Neighbors Neighbors
7877a1i 11 . . . 4 USGrph Neighbors Neighbors Neighbors
79 nbgranself2 25243 . . . . . . . . . 10 USGrph Neighbors
80 df-nel 2644 . . . . . . . . . . 11 Neighbors Neighbors
81 prid2g 4070 . . . . . . . . . . . . . 14
82813ad2ant1 1051 . . . . . . . . . . . . 13
83 eleq2 2538 . . . . . . . . . . . . 13 Neighbors Neighbors
8482, 83syl5ibrcom 230 . . . . . . . . . . . 12 Neighbors Neighbors
8584con3rr3 143 . . . . . . . . . . 11 Neighbors Neighbors
8680, 85sylbi 200 . . . . . . . . . 10 Neighbors Neighbors
8779, 86syl 17 . . . . . . . . 9 USGrph Neighbors
8887adantl 473 . . . . . . . 8 USGrph Neighbors
8988impcom 437 . . . . . . 7 USGrph Neighbors
90893adant3 1050 . . . . . 6 USGrph Neighbors
91 biorf 412 . . . . . . 7 Neighbors Neighbors Neighbors Neighbors
92 orcom 394 . . . . . . 7 Neighbors Neighbors Neighbors Neighbors
9391, 92syl6bb 269 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
9490, 93syl 17 . . . . 5 USGrph Neighbors Neighbors Neighbors
95 prid2g 4070 . . . . . . . . . . . . . 14
96953ad2ant1 1051 . . . . . . . . . . . . 13
97 eleq2 2538 . . . . . . . . . . . . 13 Neighbors Neighbors
9896, 97syl5ibrcom 230 . . . . . . . . . . . 12 Neighbors Neighbors
9998con3rr3 143 . . . . . . . . . . 11 Neighbors Neighbors
10080, 99sylbi 200 . . . . . . . . . 10 Neighbors Neighbors
10179, 100syl 17 . . . . . . . . 9 USGrph Neighbors
102101adantl 473 . . . . . . . 8 USGrph Neighbors
103102impcom 437 . . . . . . 7 USGrph Neighbors
1041033adant3 1050 . . . . . 6 USGrph Neighbors
105 biorf 412 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
106104, 105syl 17 . . . . 5 USGrph Neighbors Neighbors Neighbors
10794, 106orbi12d 724 . . . 4 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
108 prid1g 4069 . . . . . . . . . . . . . . . 16
1091083ad2ant1 1051 . . . . . . . . . . . . . . 15
110 eleq2 2538 . . . . . . . . . . . . . . 15 Neighbors Neighbors
111109, 110syl5ibrcom 230 . . . . . . . . . . . . . 14 Neighbors Neighbors
112111con3dimp 448 . . . . . . . . . . . . 13 Neighbors Neighbors
113 prid1g 4069 . . . . . . . . . . . . . . . 16
1141133ad2ant1 1051 . . . . . . . . . . . . . . 15
115 eleq2 2538 . . . . . . . . . . . . . . 15 Neighbors Neighbors
116114, 115syl5ibrcom 230 . . . . . . . . . . . . . 14 Neighbors Neighbors
117116con3dimp 448 . . . . . . . . . . . . 13 Neighbors Neighbors
118112, 117jca 541 . . . . . . . . . . . 12 Neighbors Neighbors Neighbors
119118expcom 442 . . . . . . . . . . 11 Neighbors Neighbors Neighbors
12080, 119sylbi 200 . . . . . . . . . 10 Neighbors Neighbors Neighbors
12179, 120syl 17 . . . . . . . . 9 USGrph Neighbors Neighbors
122121adantl 473 . . . . . . . 8 USGrph Neighbors Neighbors
123122impcom 437 . . . . . . 7 USGrph Neighbors Neighbors
1241233adant3 1050 . . . . . 6 USGrph Neighbors Neighbors
125 ioran 498 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
126124, 125sylibr 217 . . . . 5 USGrph Neighbors Neighbors
1271263bior1fd 1403 . . . 4 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
12878, 107, 1273bitrd 287 . . 3 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
12944, 72, 1283bitr4rd 294 . 2 USGrph Neighbors Neighbors Neighbors Neighbors
13017, 23, 1293bitr4rd 294 1 USGrph Neighbors Neighbors
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   w3o 1006   w3a 1007   wceq 1452   wcel 1904   wne 2641   wnel 2642  wrex 2757   cdif 3387  csn 3959  cpr 3961  ctp 3963  cop 3965   class class class wbr 4395  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227 This theorem is referenced by:  nb3grapr  25260
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