Theorem ax13 2047

Description: Derive ax-13 1999 from ax13v 2000 via axc9 2046. This shows that the weakening in ax13v 2000 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.)

Assertion

Ref	Expression
ax13

Proof of Theorem ax13

Step	Hyp	Ref	Expression
1		sp 1859	. . . 4
2	1	con3i 135	. . 3
3		sp 1859	. . . 4
4	3	con3i 135	. . 3
5		axc9 2046	. . 3
6	2, 4, 5	syl2im 38	. 2
7		ax13b 1805	. 2
8	6, 7	mpbir 209	1

Syntax hints:  -.wn 3  ->wi 4  A.wal 1393

This theorem is referenced by:  equvini  2087  sbequi  2116

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-12 1854  ax-13 1999

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