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Theorem ax12 2234
Description: Rederivation of axiom ax-12 1854 from ax-c15 2220, ax-c11 2218, and other older axioms. See theorem axc15 2085 for the derivation of ax-c15 2220 from ax-12 1854.

An open problem is whether we can prove this using ax-c11n 2219 instead of ax-c11 2218.

This proof uses newer axioms ax-4 1631 and ax-6 1747, 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 2215 and ax-c10 2217. (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 237 . . . . 5
21dral1-o 2233 . . . 4
3 ax-1 6 . . . . 5
43alimi 1633 . . . 4
52, 4syl6bir 229 . . 3
65a1d 25 . 2
7 ax-c5 2214 . . 3
8 ax-c15 2220 . . 3
97, 8syl7 68 . 2
106, 9pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  A.wal 1393
This theorem is referenced by:  axc11-o  2281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-11 1842  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c11 2218  ax-c15 2220  ax-c9 2221
This theorem depends on definitions:  df-bi 185  df-ex 1613
  Copyright terms: Public domain W3C validator