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

Step	Hyp	Ref	Expression
1		ax6ev 1749	. 2
2		ax12v2-o.1	. . . . 5
3		equequ2 1799	. . . . . . 7
4	3	adantl 466	. . . . . 6
5		dveeq2-o 2263	. . . . . . . . 9
6	5	imp 429	. . . . . . . 8
7		nfa1-o 2245	. . . . . . . . 9
8	3	imbi1d 317	. . . . . . . . . 10
9	8	sps-o 2238	. . . . . . . . 9
10	7, 9	albid 1885	. . . . . . . 8
11	6, 10	syl 16	. . . . . . 7
12	11	imbi2d 316	. . . . . 6
13	4, 12	imbi12d 320	. . . . 5
14	2, 13	mpbii 211	. . . 4
15	14	ex 434	. . 3
16	15	exlimdv 1724	. 2
17	1, 16	mpi 17	1

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