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Theorem trsbcVD 37337
Description: Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 36971 is trsbcVD 37337 without virtual deductions and was automatically derived from trsbcVD 37337.
 1:: 2:1: 3:1: 4:1: 5:1,2,3,4: 6:1: 7:5,6: 8:: 9:7,8: 10:: 11:10: 12:1,11: 13:9,12: 14:13: 15:14: 16:1: 17:15,16: 18:17: 19:18: 20:1: 21:19,20: 22:: 23:21,22: 24:: 25:24: 26:1,25: 27:23,26: qed:27:
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsbcVD
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem trsbcVD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 37012 . . . . . . . . . . . . . 14
2 biidd 245 . . . . . . . . . . . . . . . 16
32sbcieg 3288 . . . . . . . . . . . . . . 15
41, 3e1a 37074 . . . . . . . . . . . . . 14
5 sbcel2gv 3315 . . . . . . . . . . . . . . 15
61, 5e1a 37074 . . . . . . . . . . . . . 14
7 sbcel2gv 3315 . . . . . . . . . . . . . . 15
81, 7e1a 37074 . . . . . . . . . . . . . 14
9 imbi13 36947 . . . . . . . . . . . . . . 15
109a1i 11 . . . . . . . . . . . . . 14
111, 4, 6, 8, 10e1111 37122 . . . . . . . . . . . . 13
12 sbcim2g 36969 . . . . . . . . . . . . . 14
131, 12e1a 37074 . . . . . . . . . . . . 13
14 bibi1 334 . . . . . . . . . . . . . 14
1514biimprcd 233 . . . . . . . . . . . . 13
1611, 13, 15e11 37135 . . . . . . . . . . . 12
17 pm3.31 452 . . . . . . . . . . . . 13
18 pm3.3 451 . . . . . . . . . . . . 13
1917, 18impbii 192 . . . . . . . . . . . 12
20 bibi1 334 . . . . . . . . . . . . 13
2120biimprd 231 . . . . . . . . . . . 12
2216, 19, 21e10 37141 . . . . . . . . . . 11
23 pm3.31 452 . . . . . . . . . . . . . 14
24 pm3.3 451 . . . . . . . . . . . . . 14
2523, 24impbii 192 . . . . . . . . . . . . 13
2625ax-gen 1677 . . . . . . . . . . . 12
27 sbcbi 36970 . . . . . . . . . . . 12
281, 26, 27e10 37141 . . . . . . . . . . 11
29 bitr3 36937 . . . . . . . . . . . 12
3029com12 31 . . . . . . . . . . 11
3122, 28, 30e11 37135 . . . . . . . . . 10
3231gen11 37063 . . . . . . . . 9
33 albi 1698 . . . . . . . . 9
3432, 33e1a 37074 . . . . . . . 8
35 sbcalgOLD 36973 . . . . . . . . 9
361, 35e1a 37074 . . . . . . . 8
37 bibi1 334 . . . . . . . . 9
3837biimprcd 233 . . . . . . . 8
3934, 36, 38e11 37135 . . . . . . 7
4039gen11 37063 . . . . . 6
41 albi 1698 . . . . . 6
4240, 41e1a 37074 . . . . 5
43 sbcalgOLD 36973 . . . . . 6
441, 43e1a 37074 . . . . 5
45 bibi1 334 . . . . . 6
4645biimprcd 233 . . . . 5
4742, 44, 46e11 37135 . . . 4
48 dftr2 4492 . . . 4
49 biantr 945 . . . . 5
5049ex 441 . . . 4
5147, 48, 50e10 37141 . . 3
52 dftr2 4492 . . . . 5
5352ax-gen 1677 . . . 4
54 sbcbi 36970 . . . 4
551, 53, 54e10 37141 . . 3
56 bibi1 334 . . . 4
5756biimprcd 233 . . 3
5851, 55, 57e11 37135 . 2
5958in1 37009 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450   wceq 1452   wcel 1904  wsbc 3255   wtr 4490 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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491  df-vd1 37008 This theorem is referenced by: (None)
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