Theorem ax13fromc9 2235

Description: Derive ax-13 1999 from ax-c9 2221 and other older axioms.

This proof uses newer axioms ax-4 1631 and ax-6 1747, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 2215 and ax-c10 2217. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13fromc9

Proof of Theorem ax13fromc9

Step	Hyp	Ref	Expression
1		ax-c5 2214	. . . . . 6
2	1	con3i 135	. . . . 5
3	2	adantr 465	. . . 4
4		equtrr 1797	. . . . . . . 8
5	4	equcoms 1795	. . . . . . 7
6	5	con3rr3 136	. . . . . 6
7	6	imp 429	. . . . 5
8		ax-c5 2214	. . . . 5
9	7, 8	nsyl 121	. . . 4
10		ax-c9 2221	. . . 4
11	3, 9, 10	sylc 60	. . 3
12	11	ex 434	. 2
13	12	pm2.43d 48	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393

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-c5 2214  ax-c9 2221

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