Theorem ax13 2051

Description: Derive ax-13 2004 from ax13v 2005 via axc9 2050. This shows that the weakening in ax13v 2005 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	|- ( -. x = y -> ( y = z -> A. x y = z ) )

Proof of Theorem ax13

Step	Hyp	Ref	Expression
1		sp 1864	. . . 4 |- ( A. x x = y -> x = y )
2	1	con3i 135	. . 3 |- ( -. x = y -> -. A. x x = y )
3		sp 1864	. . . 4 |- ( A. x x = z -> x = z )
4	3	con3i 135	. . 3 |- ( -. x = z -> -. A. x x = z )
5		axc9 2050	. . 3 |- ( -. A. x x = y -> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) )
6	2, 4, 5	syl2im 38	. 2 |- ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) )
7		ax13b 1810	. 2 |- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
8	6, 7	mpbir 209	1 |- ( -. x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622

This theorem is referenced by:  equvini  2091  sbequi  2118