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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax12fromc15 | Structured version Visualization version Unicode version |
Description: Rederivation of axiom ax-12 2047 from ax-c15 34174, ax-c11 34172 (used through
dral1-o 34189), and other older axioms. See theorem axc15 2303 for the
derivation of ax-c15 34174 from ax-12 2047.
An open problem is whether we can prove this using ax-c11n 34173 instead of ax-c11 34172. This proof uses newer axioms ax-4 1737 and ax-6 1888, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 34169 and ax-c10 34171. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12fromc15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 252 | . . . . 5 | |
2 | 1 | dral1-o 34189 | . . . 4 |
3 | ax-1 6 | . . . . 5 | |
4 | 3 | alimi 1739 | . . . 4 |
5 | 2, 4 | syl6bir 244 | . . 3 |
6 | 5 | a1d 25 | . 2 |
7 | ax-c5 34168 | . . 3 | |
8 | ax-c15 34174 | . . 3 | |
9 | 7, 8 | syl7 74 | . 2 |
10 | 6, 9 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-11 2034 ax-c5 34168 ax-c4 34169 ax-c7 34170 ax-c10 34171 ax-c11 34172 ax-c15 34174 ax-c9 34175 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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