Theorem ax12inda 2278

Description: Induction step for constructing a substitution instance of ax-c15 2220 without using ax-c15 2220. Quantification case. (When and are distinct, ax12inda2 2277 may be used instead to avoid the dummy variable in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12inda.1

Assertion

Ref	Expression
ax12inda

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem ax12inda

Step	Hyp	Ref	Expression
1		ax6ev 1749	. . 3
2		ax12inda.1	. . . . . . 7
3	2	ax12inda2 2277	. . . . . 6
4		dveeq2-o 2263	. . . . . . . . 9
5	4	imp 429	. . . . . . . 8
6		hba1-o 2228	. . . . . . . . . 10
7		equequ2 1799	. . . . . . . . . . 11
8	7	sps-o 2238	. . . . . . . . . 10
9	6, 8	albidh 1675	. . . . . . . . 9
10	9	notbid 294	. . . . . . . 8
11	5, 10	syl 16	. . . . . . 7
12	7	adantl 466	. . . . . . . 8
13	8	imbi1d 317	. . . . . . . . . . 11
14	6, 13	albidh 1675	. . . . . . . . . 10
15	5, 14	syl 16	. . . . . . . . 9
16	15	imbi2d 316	. . . . . . . 8
17	12, 16	imbi12d 320	. . . . . . 7
18	11, 17	imbi12d 320	. . . . . 6
19	3, 18	mpbii 211	. . . . 5
20	19	ex 434	. . . 4
21	20	exlimdv 1724	. . 3
22	1, 21	mpi 17	. 2
23	22	pm2.43i 47	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612

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  ax-c16 2223

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617