Theorem ax13fromc9 34191

Description: Derive ax-13 2246 from ax-c9 34175 and other older axioms.

This proof uses newer axioms ax-4 1737 and ax-6 1888, 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 34169 and ax-c10 34171. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13fromc9	|- ( -. x = y -> ( y = z -> A. x y = z ) )

Proof of Theorem ax13fromc9

Step	Hyp	Ref	Expression
1		ax-c5 34168	. . . 4 |- ( A. x x = y -> x = y )
2	1	con3i 150	. . 3 |- ( -. x = y -> -. A. x x = y )
3		ax-c5 34168	. . . 4 |- ( A. x x = z -> x = z )
4	3	con3i 150	. . 3 |- ( -. x = z -> -. A. x x = z )
5		ax-c9 34175	. . 3 |- ( -. A. x x = y -> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) )
6	2, 4, 5	syl2im 40	. 2 |- ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) )
7		ax13b 1964	. 2 |- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
8	6, 7	mpbir 221	1 |- ( -. x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-c5 34168  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)