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Mirrors > Home > ILE Home > Th. List > ax16 | Unicode version |
Description: Theorem showing that ax-16 1786 is redundant if ax-17 1506 is included in the
axiom system. The important part of the proof is provided by aev 1784.
See ax16ALT 1831 for an alternate proof that does not require ax-10 1483 or ax-12 1489. This theorem should not be referenced in any proof. Instead, use ax-16 1786 below so that theorems needing ax-16 1786 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Ref | Expression |
---|---|
ax16 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 1784 | . 2 | |
2 | ax-17 1506 | . . . 4 | |
3 | sbequ12 1744 | . . . . 5 | |
4 | 3 | biimpcd 158 | . . . 4 |
5 | 2, 4 | alimdh 1443 | . . 3 |
6 | 2 | hbsb3 1780 | . . . 4 |
7 | stdpc7 1743 | . . . 4 | |
8 | 6, 2, 7 | cbv3h 1721 | . . 3 |
9 | 5, 8 | syl6com 35 | . 2 |
10 | 1, 9 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1329 wsb 1735 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: dveeq2 1787 dveeq2or 1788 a16g 1836 exists2 2094 |
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