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Theorem ax12v2-o 32232
 Description: Recovery of ax-c15 32173 from ax12v 1908 without using ax-c15 32173. The hypothesis is even weaker than ax12v 1908, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-c15 32173. (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:   ,   ,   ,
Allowed substitution hints:   (,)

Proof of Theorem ax12v2-o
StepHypRef Expression
1 ax6ev 1799 . 2
2 ax12v2-o.1 . . . . 5
3 equequ2 1851 . . . . . . 7
43adantl 467 . . . . . 6
5 dveeq2-o 32216 . . . . . . . . 9
65imp 430 . . . . . . . 8
7 nfa1-o 32198 . . . . . . . . 9
83imbi1d 318 . . . . . . . . . 10
98sps-o 32191 . . . . . . . . 9
107, 9albid 1938 . . . . . . . 8
116, 10syl 17 . . . . . . 7
1211imbi2d 317 . . . . . 6
134, 12imbi12d 321 . . . . 5
142, 13mpbii 214 . . . 4
1514ex 435 . . 3
1615exlimdv 1771 . 2
171, 16mpi 21 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370  wal 1435  wex 1659 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-c5 32167  ax-c4 32168  ax-c7 32169  ax-c11 32171  ax-c9 32174 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664 This theorem is referenced by:  ax12a2-o  32233
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