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Mirrors > Home > ILE Home > Th. List > poeq1 | Unicode version |
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
poeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 3931 | . . . . . 6 | |
2 | 1 | notbid 656 | . . . . 5 |
3 | breq 3931 | . . . . . . 7 | |
4 | breq 3931 | . . . . . . 7 | |
5 | 3, 4 | anbi12d 464 | . . . . . 6 |
6 | breq 3931 | . . . . . 6 | |
7 | 5, 6 | imbi12d 233 | . . . . 5 |
8 | 2, 7 | anbi12d 464 | . . . 4 |
9 | 8 | ralbidv 2437 | . . 3 |
10 | 9 | 2ralbidv 2459 | . 2 |
11 | df-po 4218 | . 2 | |
12 | df-po 4218 | . 2 | |
13 | 10, 11, 12 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wral 2416 class class class wbr 3929 wpo 4216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2132 df-clel 2135 df-ral 2421 df-br 3930 df-po 4218 |
This theorem is referenced by: soeq1 4237 |
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