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Theorem ax12v2 2083
Description: Recovery of ax-c15 2220 from ax12v 1855. This proof uses ax-c11n 2219 and ax-12 1854. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
Hypothesis
Ref Expression
ax12v2.1
Assertion
Ref Expression
ax12v2
Distinct variable groups:   ,   ,   ,

Proof of Theorem ax12v2
StepHypRef Expression
1 ax6ev 1749 . 2
2 dveeq2 2042 . . . 4
3 ax12v2.1 . . . . 5
4 equequ2 1799 . . . . . . 7
54sps 1865 . . . . . 6
6 nfa1 1897 . . . . . . . 8
75imbi1d 317 . . . . . . . 8
86, 7albid 1885 . . . . . . 7
98imbi2d 316 . . . . . 6
105, 9imbi12d 320 . . . . 5
113, 10mpbii 211 . . . 4
122, 11syl6 33 . . 3
1312exlimdv 1724 . 2
141, 13mpi 17 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  A.wal 1393  E.wex 1612
This theorem is referenced by:  ax12a2  2084
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-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617
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