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Mirrors > Home > MPE Home > Th. List > Mathboxes > trsbcVD | Structured version Visualization version Unicode version |
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 38750 is trsbcVD 39113 without virtual deductions and was automatically
derived from trsbcVD 39113.
|
Ref | Expression |
---|---|
trsbcVD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 38790 | . . . . . . . . . . . . . 14 | |
2 | biidd 252 | . . . . . . . . . . . . . . . 16 | |
3 | 2 | sbcieg 3468 | . . . . . . . . . . . . . . 15 |
4 | 1, 3 | e1a 38852 | . . . . . . . . . . . . . 14 |
5 | sbcel2gv 3496 | . . . . . . . . . . . . . . 15 | |
6 | 1, 5 | e1a 38852 | . . . . . . . . . . . . . 14 |
7 | sbcel2gv 3496 | . . . . . . . . . . . . . . 15 | |
8 | 1, 7 | e1a 38852 | . . . . . . . . . . . . . 14 |
9 | imbi13 38726 | . . . . . . . . . . . . . . 15 | |
10 | 9 | a1i 11 | . . . . . . . . . . . . . 14 |
11 | 1, 4, 6, 8, 10 | e1111 38900 | . . . . . . . . . . . . 13 |
12 | sbcim2g 38748 | . . . . . . . . . . . . . 14 | |
13 | 1, 12 | e1a 38852 | . . . . . . . . . . . . 13 |
14 | bibi1 341 | . . . . . . . . . . . . . 14 | |
15 | 14 | biimprcd 240 | . . . . . . . . . . . . 13 |
16 | 11, 13, 15 | e11 38913 | . . . . . . . . . . . 12 |
17 | pm3.31 461 | . . . . . . . . . . . . 13 | |
18 | pm3.3 460 | . . . . . . . . . . . . 13 | |
19 | 17, 18 | impbii 199 | . . . . . . . . . . . 12 |
20 | bibi1 341 | . . . . . . . . . . . . 13 | |
21 | 20 | biimprd 238 | . . . . . . . . . . . 12 |
22 | 16, 19, 21 | e10 38919 | . . . . . . . . . . 11 |
23 | pm3.31 461 | . . . . . . . . . . . . . 14 | |
24 | pm3.3 460 | . . . . . . . . . . . . . 14 | |
25 | 23, 24 | impbii 199 | . . . . . . . . . . . . 13 |
26 | 25 | ax-gen 1722 | . . . . . . . . . . . 12 |
27 | sbcbi 38749 | . . . . . . . . . . . 12 | |
28 | 1, 26, 27 | e10 38919 | . . . . . . . . . . 11 |
29 | bitr3 342 | . . . . . . . . . . . 12 | |
30 | 29 | com12 32 | . . . . . . . . . . 11 |
31 | 22, 28, 30 | e11 38913 | . . . . . . . . . 10 |
32 | 31 | gen11 38841 | . . . . . . . . 9 |
33 | albi 1746 | . . . . . . . . 9 | |
34 | 32, 33 | e1a 38852 | . . . . . . . 8 |
35 | sbcalgOLD 38752 | . . . . . . . . 9 | |
36 | 1, 35 | e1a 38852 | . . . . . . . 8 |
37 | bibi1 341 | . . . . . . . . 9 | |
38 | 37 | biimprcd 240 | . . . . . . . 8 |
39 | 34, 36, 38 | e11 38913 | . . . . . . 7 |
40 | 39 | gen11 38841 | . . . . . 6 |
41 | albi 1746 | . . . . . 6 | |
42 | 40, 41 | e1a 38852 | . . . . 5 |
43 | sbcalgOLD 38752 | . . . . . 6 | |
44 | 1, 43 | e1a 38852 | . . . . 5 |
45 | bibi1 341 | . . . . . 6 | |
46 | 45 | biimprcd 240 | . . . . 5 |
47 | 42, 44, 46 | e11 38913 | . . . 4 |
48 | dftr2 4754 | . . . 4 | |
49 | biantr 972 | . . . . 5 | |
50 | 49 | ex 450 | . . . 4 |
51 | 47, 48, 50 | e10 38919 | . . 3 |
52 | dftr2 4754 | . . . . 5 | |
53 | 52 | ax-gen 1722 | . . . 4 |
54 | sbcbi 38749 | . . . 4 | |
55 | 1, 53, 54 | e10 38919 | . . 3 |
56 | bibi1 341 | . . . 4 | |
57 | 56 | biimprcd 240 | . . 3 |
58 | 51, 55, 57 | e11 38913 | . 2 |
59 | 58 | in1 38787 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wcel 1990 wsbc 3435 wtr 4752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 df-in 3581 df-ss 3588 df-uni 4437 df-tr 4753 df-vd1 38786 |
This theorem is referenced by: (None) |
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