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Mirrors > Home > ILE Home > Th. List > aev | Unicode version |
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1787. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
aev |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbae 1697 | . 2 | |
2 | hbae 1697 | . . . 4 | |
3 | ax-8 1483 | . . . . 5 | |
4 | 3 | spimv 1784 | . . . 4 |
5 | 2, 4 | alrimih 1446 | . . 3 |
6 | ax-8 1483 | . . . . . . . 8 | |
7 | equcomi 1681 | . . . . . . . 8 | |
8 | 6, 7 | syl6 33 | . . . . . . 7 |
9 | 8 | spimv 1784 | . . . . . 6 |
10 | 9 | alequcoms 1497 | . . . . 5 |
11 | 10 | a5i 1523 | . . . 4 |
12 | hbae 1697 | . . . . 5 | |
13 | ax-8 1483 | . . . . . 6 | |
14 | 13 | spimv 1784 | . . . . 5 |
15 | 12, 14 | alrimih 1446 | . . . 4 |
16 | alequcom 1496 | . . . 4 | |
17 | 11, 15, 16 | 3syl 17 | . . 3 |
18 | ax-8 1483 | . . . 4 | |
19 | 18 | spimv 1784 | . . 3 |
20 | 5, 17, 19 | 3syl 17 | . 2 |
21 | 1, 20 | alrimih 1446 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1330 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: ax16 1786 a16g 1837 |
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