Theorem ax13 2249

Description: Derive ax-13 2246 from ax13v 2247 and Tarski's FOL. This shows that the weakening in ax13v 2247 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage (Revised by Wolf Lammen, 2-Jun-2021.)

Assertion

Ref	Expression
ax13	|- ( -. x = y -> ( y = z -> A. x y = z ) )

Proof of Theorem ax13

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 1959	. . . 4 |- ( y = z <-> E. w ( w = y /\ w = z ) )
2		ax13lem1 2248	. . . . . . . . 9 |- ( -. x = y -> ( w = y -> A. x w = y ) )
3	2	imp 445	. . . . . . . 8 |- ( ( -. x = y /\ w = y ) -> A. x w = y )
4		ax13lem1 2248	. . . . . . . . 9 |- ( -. x = z -> ( w = z -> A. x w = z ) )
5	4	imp 445	. . . . . . . 8 |- ( ( -. x = z /\ w = z ) -> A. x w = z )
6		ax7v1 1937	. . . . . . . . . 10 |- ( w = y -> ( w = z -> y = z ) )
7	6	imp 445	. . . . . . . . 9 |- ( ( w = y /\ w = z ) -> y = z )
8	7	alanimi 1744	. . . . . . . 8 |- ( ( A. x w = y /\ A. x w = z ) -> A. x y = z )
9	3, 5, 8	syl2an 494	. . . . . . 7 |- ( ( ( -. x = y /\ w = y ) /\ ( -. x = z /\ w = z ) ) -> A. x y = z )
10	9	an4s 869	. . . . . 6 |- ( ( ( -. x = y /\ -. x = z ) /\ ( w = y /\ w = z ) ) -> A. x y = z )
11	10	ex 450	. . . . 5 |- ( ( -. x = y /\ -. x = z ) -> ( ( w = y /\ w = z ) -> A. x y = z ) )
12	11	exlimdv 1861	. . . 4 |- ( ( -. x = y /\ -. x = z ) -> ( E. w ( w = y /\ w = z ) -> A. x y = z ) )
13	1, 12	syl5bi 232	. . 3 |- ( ( -. x = y /\ -. x = z ) -> ( y = z -> A. x y = z ) )
14	13	ex 450	. 2 |- ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) )
15		ax13b 1964	. 2 |- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
16	14, 15	mpbir 221	1 |- ( -. x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

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-13 2246

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

This theorem is referenced by:  equvini  2346  sbequi  2375