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Theorem trsbcVD 28990
Description: Formula-building inference rule for class substitution, substituting a class variable for the set 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 O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 28626 is trsbcVD 28990 without virtual deductions and was automatically derived from trsbcVD 28990.
 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 28666 . . . . . . . . . . . . . 14
2 biidd 230 . . . . . . . . . . . . . . . 16
32sbcieg 3194 . . . . . . . . . . . . . . 15
41, 3e1_ 28729 . . . . . . . . . . . . . 14
5 sbcel2gv 3222 . . . . . . . . . . . . . . 15
61, 5e1_ 28729 . . . . . . . . . . . . . 14
7 sbcel2gv 3222 . . . . . . . . . . . . . . 15
81, 7e1_ 28729 . . . . . . . . . . . . . 14
9 imbi13 28605 . . . . . . . . . . . . . . 15
109a1i 11 . . . . . . . . . . . . . 14
111, 4, 6, 8, 10e1111 28777 . . . . . . . . . . . . 13
12 sbcim2g 28624 . . . . . . . . . . . . . 14
131, 12e1_ 28729 . . . . . . . . . . . . 13
14 bibi1 319 . . . . . . . . . . . . . 14
1514biimprcd 218 . . . . . . . . . . . . 13
1611, 13, 15e11 28790 . . . . . . . . . . . 12
17 pm3.31 434 . . . . . . . . . . . . 13
18 pm3.3 433 . . . . . . . . . . . . 13
1917, 18impbii 182 . . . . . . . . . . . 12
20 bibi1 319 . . . . . . . . . . . . 13
2120biimprd 216 . . . . . . . . . . . 12
2216, 19, 21e10 28796 . . . . . . . . . . 11
23 pm3.31 434 . . . . . . . . . . . . . 14
24 pm3.3 433 . . . . . . . . . . . . . 14
2523, 24impbii 182 . . . . . . . . . . . . 13
2625ax-gen 1556 . . . . . . . . . . . 12
27 sbcbi 28625 . . . . . . . . . . . 12
281, 26, 27e10 28796 . . . . . . . . . . 11
29 bitr3 28594 . . . . . . . . . . . 12
3029com12 30 . . . . . . . . . . 11
3122, 28, 30e11 28790 . . . . . . . . . 10
3231gen11 28718 . . . . . . . . 9
33 albi 1574 . . . . . . . . 9
3432, 33e1_ 28729 . . . . . . . 8
35 sbcalg 3210 . . . . . . . . 9
361, 35e1_ 28729 . . . . . . . 8
37 bibi1 319 . . . . . . . . 9
3837biimprcd 218 . . . . . . . 8
3934, 36, 38e11 28790 . . . . . . 7
4039gen11 28718 . . . . . 6
41 albi 1574 . . . . . 6
4240, 41e1_ 28729 . . . . 5
43 sbcalg 3210 . . . . . 6
441, 43e1_ 28729 . . . . 5
45 bibi1 319 . . . . . 6
4645biimprcd 218 . . . . 5
4742, 44, 46e11 28790 . . . 4
48 dftr2 4305 . . . 4
49 biantr 899 . . . . 5
5049ex 425 . . . 4
5147, 48, 50e10 28796 . . 3
52 dftr2 4305 . . . . 5
5352ax-gen 1556 . . . 4
54 sbcbi 28625 . . . 4
551, 53, 54e10 28796 . . 3
56 bibi1 319 . . . 4
5756biimprcd 218 . . 3
5851, 55, 57e11 28790 . 2
5958in1 28663 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wal 1550   wceq 1653   wcel 1726  wsbc 3162   wtr 4303 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163  df-in 3328  df-ss 3335  df-uni 4017  df-tr 4304  df-vd1 28662
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