MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12v2-o Unicode version

Theorem ax12v2-o 2279
Description: Recovery of ax-c15 2220 from ax12v 1855 without using ax-c15 2220. The hypothesis is even weaker than ax12v 1855, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-c15 2220. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12v2-o.1
Assertion
Ref Expression
ax12v2-o
Distinct variable groups:   ,   ,   ,

Proof of Theorem ax12v2-o
StepHypRef Expression
1 ax6ev 1749 . 2
2 ax12v2-o.1 . . . . 5
3 equequ2 1799 . . . . . . 7
43adantl 466 . . . . . 6
5 dveeq2-o 2263 . . . . . . . . 9
65imp 429 . . . . . . . 8
7 nfa1-o 2245 . . . . . . . . 9
83imbi1d 317 . . . . . . . . . 10
98sps-o 2238 . . . . . . . . 9
107, 9albid 1885 . . . . . . . 8
116, 10syl 16 . . . . . . 7
1211imbi2d 316 . . . . . 6
134, 12imbi12d 320 . . . . 5
142, 13mpbii 211 . . . 4
1514ex 434 . . 3
1615exlimdv 1724 . 2
171, 16mpi 17 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612
This theorem is referenced by:  ax12a2-o  2280
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c11 2218  ax-c9 2221
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
  Copyright terms: Public domain W3C validator