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| Description: Rederivation of axiom ax-12 1009 from two shorter formulas, without using ax-12 1009. See a12lem1 1418 and a12lem2 1419 for the proofs of the hypotheses (using ax-12 1009). This is the only known breakdown of ax-12 1009 into shorter formulas. See a12studyALT 1421 for an alternate proof. Note that the proof depends on ax-11o 1260, whose proof ax11o 1259 depends on ax-12 1009, meaning that we would have to replace ax-11 1008 with ax-11o 1260 in an axiomatization that uses the hypotheses in place of ax-12 1009. Whether this can be avoided is an open problem. |
| Ref | Expression |
|---|---|
| a12study.1 |
|
| a12study.2 |
|
| Ref | Expression |
|---|---|
| a12study |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hbn1 1056 |
. . . . 5
| |
| 2 | hbn1 1056 |
. . . . 5
| |
| 3 | 1, 2 | hban 1050 |
. . . 4
|
| 4 | hba1 1044 |
. . . 4
| |
| 5 | ax-11o 1260 |
. . . . . . 7
| |
| 6 | equid1 1311 |
. . . . . . . 8
| |
| 7 | ax-8 1005 |
. . . . . . . 8
| |
| 8 | 6, 7 | mpi 44 |
. . . . . . 7
|
| 9 | 5, 8 | syl5 21 |
. . . . . 6
|
| 10 | 9 | imp3a 368 |
. . . . 5
|
| 11 | hba1 1044 |
. . . . . 6
| |
| 12 | a12study.1 |
. . . . . 6
| |
| 13 | 2, 11, 12 | 19.21ad 1100 |
. . . . 5
|
| 14 | 10, 13 | sylan9 479 |
. . . 4
|
| 15 | 3, 4, 14 | 19.23ad 1107 |
. . 3
|
| 16 | 15 | ex 380 |
. 2
|
| 17 | imnan 249 |
. . . . . . 7
| |
| 18 | equid1 1311 |
. . . . . . . . 9
| |
| 19 | ax-8 1005 |
. . . . . . . . 9
| |
| 20 | 18, 19 | mpi 44 |
. . . . . . . 8
|
| 21 | 20 | imim1i 16 |
. . . . . . 7
|
| 22 | 17, 21 | sylbir 208 |
. . . . . 6
|
| 23 | 22 | 19.20i 1033 |
. . . . 5
|
| 24 | a12study.2 |
. . . . 5
| |
| 25 | 23, 24 | syl 10 |
. . . 4
|
| 26 | 25 | con2i 102 |
. . 3
|
| 27 | df-ex 1022 |
. . 3
| |
| 28 | 26, 27 | sylibr 207 |
. 2
|
| 29 | 16, 28 | syl7 23 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-gen 1004 ax-8 1005 ax-9 1006 ax-17 1012 ax-4 1014 ax-5o 1016 ax-6o 1019 ax-11o 1260 |
| This theorem depends on definitions: df-bi 154 df-an 232 df-ex 1022 |