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| Mirrors > Home > MPE Home > Th. List > nic-luk1 | Structured version Visualization version Unicode version | ||
| Description: Proof of luk-1 1580 from nic-ax 1598 and nic-mp 1596 (and definitions nic-dfim 1594 and nic-dfneg 1595). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by theorems ax1 1591, ax2 1592, and ax3 1593. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nic-luk1 |
      
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nic-dfim 1594 |
. . . 4
![-/\](barwedge.gif "-/\"){kind=small}
| |
| 2 | 1 | nic-bi2 1614 |
. . 3
|
| 3 | nic-ax 1598 |
. . . . . . 7
| |
| 4 | 3 | nic-isw2 1606 |
. . . . . 6
|
| 5 | 4 | nic-idel 1609 |
. . . . 5
|
| 6 | nic-dfim 1594 |
. . . . . . . . 9
| |
| 7 | 6 | nic-bi1 1613 |
. . . . . . . 8
|
| 8 | 7 | nic-idbl 1611 |
. . . . . . 7
|
| 9 | 8 | nic-imp 1600 |
. . . . . 6
|
| 10 | nic-dfim 1594 |
. . . . . . . . 9
| |
| 11 | 10 | nic-bi2 1614 |
. . . . . . . 8
|
| 12 | nic-swap 1604 |
. . . . . . . 8
| |
| 13 | 11, 12 | nic-ich 1610 |
. . . . . . 7
|
| 14 | 13 | nic-imp 1600 |
. . . . . 6
|
| 15 | 9, 14 | nic-ich 1610 |
. . . . 5
|
| 16 | 5, 15 | nic-ich 1610 |
. . . 4
|
| 17 | nic-dfim 1594 |
. . . . 5
| |
| 18 | 17 | nic-bi1 1613 |
. . . 4
|
| 19 | 16, 18 | nic-ich 1610 |
. . 3
|
| 20 | 2, 19 | nic-ich 1610 |
. 2
|
| 21 | nic-dfim 1594 |
. . 3
| |
| 22 | 21 | nic-bi1 1613 |
. 2
|
| 23 | 20, 22 | nic-mp 1596 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wnan 1447 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-nan 1448 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |