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Theorem trsbcVD 28346
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 27984 is trsbcVD 28346 without virtual deductions and was automatically derived from trsbcVD 28346.
 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 28022 . . . . . . . . . . . . . 14
2 biidd 229 . . . . . . . . . . . . . . . 16
32sbcieg 3150 . . . . . . . . . . . . . . 15
41, 3e1_ 28085 . . . . . . . . . . . . . 14
5 sbcel2gv 3178 . . . . . . . . . . . . . . 15
61, 5e1_ 28085 . . . . . . . . . . . . . 14
7 sbcel2gv 3178 . . . . . . . . . . . . . . 15
81, 7e1_ 28085 . . . . . . . . . . . . . 14
9 imbi13 27963 . . . . . . . . . . . . . . 15
109a1i 11 . . . . . . . . . . . . . 14
111, 4, 6, 8, 10e1111 28133 . . . . . . . . . . . . 13
12 sbcim2g 27982 . . . . . . . . . . . . . 14
131, 12e1_ 28085 . . . . . . . . . . . . 13
14 bibi1 318 . . . . . . . . . . . . . 14
1514biimprcd 217 . . . . . . . . . . . . 13
1611, 13, 15e11 28146 . . . . . . . . . . . 12
17 pm3.31 433 . . . . . . . . . . . . 13
18 pm3.3 432 . . . . . . . . . . . . 13
1917, 18impbii 181 . . . . . . . . . . . 12
20 bibi1 318 . . . . . . . . . . . . 13
2120biimprd 215 . . . . . . . . . . . 12
2216, 19, 21e10 28152 . . . . . . . . . . 11
23 pm3.31 433 . . . . . . . . . . . . . 14
24 pm3.3 432 . . . . . . . . . . . . . 14
2523, 24impbii 181 . . . . . . . . . . . . 13
2625ax-gen 1552 . . . . . . . . . . . 12
27 sbcbi 27983 . . . . . . . . . . . 12
281, 26, 27e10 28152 . . . . . . . . . . 11
29 bitr3 27952 . . . . . . . . . . . 12
3029com12 29 . . . . . . . . . . 11
3122, 28, 30e11 28146 . . . . . . . . . 10
3231gen11 28074 . . . . . . . . 9
33 albi 1570 . . . . . . . . 9
3432, 33e1_ 28085 . . . . . . . 8
35 sbcalg 3166 . . . . . . . . 9
361, 35e1_ 28085 . . . . . . . 8
37 bibi1 318 . . . . . . . . 9
3837biimprcd 217 . . . . . . . 8
3934, 36, 38e11 28146 . . . . . . 7
4039gen11 28074 . . . . . 6
41 albi 1570 . . . . . 6
4240, 41e1_ 28085 . . . . 5
43 sbcalg 3166 . . . . . 6
441, 43e1_ 28085 . . . . 5
45 bibi1 318 . . . . . 6
4645biimprcd 217 . . . . 5
4742, 44, 46e11 28146 . . . 4
48 dftr2 4259 . . . 4
49 biantr 898 . . . . 5
5049ex 424 . . . 4
5147, 48, 50e10 28152 . . 3
52 dftr2 4259 . . . . 5
5352ax-gen 1552 . . . 4
54 sbcbi 27983 . . . 4
551, 53, 54e10 28152 . . 3
56 bibi1 318 . . . 4
5756biimprcd 217 . . 3
5851, 55, 57e11 28146 . 2
5958in1 28019 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1546   wceq 1649   wcel 1721  wsbc 3118   wtr 4257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-v 2915  df-sbc 3119  df-in 3284  df-ss 3291  df-uni 3972  df-tr 4258  df-vd1 28018
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