Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax12 Structured version   Unicode version

Theorem ax12 32400
 Description: Rederivation of axiom ax-12 1906 from ax-c15 32386, ax-c11 32384, and other older axioms. See theorem axc15 2141 for the derivation of ax-c15 32386 from ax-12 1906. An open problem is whether we can prove this using ax-c11n 32385 instead of ax-c11 32384. This proof uses newer axioms ax-4 1679 and ax-6 1795, 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 32381 and ax-c10 32383. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax12

Proof of Theorem ax12
StepHypRef Expression
1 biidd 241 . . . . 5
21dral1-o 32399 . . . 4
3 ax-1 6 . . . . 5
43alimi 1681 . . . 4
52, 4syl6bir 233 . . 3
65a1d 27 . 2
7 ax-c5 32380 . . 3
8 ax-c15 32386 . . 3
97, 8syl7 71 . 2
106, 9pm2.61i 168 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4  wal 1436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-11 1893  ax-c5 32380  ax-c4 32381  ax-c7 32382  ax-c11 32384  ax-c15 32386  ax-c9 32387 This theorem depends on definitions:  df-bi 189  df-ex 1661 This theorem is referenced by:  axc11-o  32447
 Copyright terms: Public domain W3C validator