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Mirrors > Home > MPE Home > Th. List > Mathboxes > trsbc | 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. trsbc 38750 is trsbcVD 39113 without virtual deductions and was automatically derived from trsbcVD 39113 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
trsbc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcal 3485 | . . 3 | |
2 | sbcal 3485 | . . . . 5 | |
3 | sbcim2g 38748 | . . . . . . . 8 | |
4 | sbcg 3503 | . . . . . . . . 9 | |
5 | sbcel2gv 3496 | . . . . . . . . 9 | |
6 | sbcel2gv 3496 | . . . . . . . . 9 | |
7 | imbi13 38726 | . . . . . . . . 9 | |
8 | 4, 5, 6, 7 | syl3c 66 | . . . . . . . 8 |
9 | 3, 8 | bitrd 268 | . . . . . . 7 |
10 | pm3.31 461 | . . . . . . . . 9 | |
11 | pm3.3 460 | . . . . . . . . 9 | |
12 | 10, 11 | impbii 199 | . . . . . . . 8 |
13 | 12 | sbcbii 3491 | . . . . . . 7 |
14 | pm3.31 461 | . . . . . . . 8 | |
15 | pm3.3 460 | . . . . . . . 8 | |
16 | 14, 15 | impbii 199 | . . . . . . 7 |
17 | 9, 13, 16 | 3bitr3g 302 | . . . . . 6 |
18 | 17 | albidv 1849 | . . . . 5 |
19 | 2, 18 | syl5bb 272 | . . . 4 |
20 | 19 | albidv 1849 | . . 3 |
21 | 1, 20 | syl5bb 272 | . 2 |
22 | dftr2 4754 | . . 3 | |
23 | 22 | sbcbii 3491 | . 2 |
24 | dftr2 4754 | . 2 | |
25 | 21, 23, 24 | 3bitr4g 303 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 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-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 |
This theorem is referenced by: truniALT 38751 truniALTVD 39114 trintALTVD 39116 trintALT 39117 |
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