Theorem axc11n-16 2268

Description: This theorem shows that, given ax-c16 2223, we can derive a version of ax-c11n 2219. However, it is weaker than ax-c11n 2219 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11n-16

Distinct variable group:   ,

Proof of Theorem axc11n-16

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-c16 2223	. . . 4
2	1	alrimiv 1719	. . 3
3	2	axc4i-o 2229	. 2
4		equequ1 1798	. . . . . 6
5	4	cbvalv 2023	. . . . . . 7
6	5	a1i 11	. . . . . 6
7	4, 6	imbi12d 320	. . . . 5
8	7	albidv 1713	. . . 4
9	8	cbvalv 2023	. . 3
10	9	biimpi 194	. 2
11		nfa1-o 2245	. . . . . . 7
12	11	19.23 1910	. . . . . 6
13	12	albii 1640	. . . . 5
14		ax6ev 1749	. . . . . . . 8
15		pm2.27 39	. . . . . . . 8
16	14, 15	ax-mp 5	. . . . . . 7
17	16	alimi 1633	. . . . . 6
18		equequ2 1799	. . . . . . . . 9
19	18	spv 2011	. . . . . . . 8
20	19	sps-o 2238	. . . . . . 7
21	20	alcoms 1843	. . . . . 6
22	17, 21	syl 16	. . . . 5
23	13, 22	sylbi 195	. . . 4
24	23	alcoms 1843	. . 3
25	24	axc4i-o 2229	. 2
26	3, 10, 25	3syl 20	1

Syntax hints:  ->wi 4  <->wb 184  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-c16 2223

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