Theorem axc9lem2OLD 2144

Description: Obsolete proof of axc9lem2 2143 as of 18-Oct-2020. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc9lem2OLD	|- ( -. x = y -> ( E. x z = y -> z = y ) )

Distinct variable group:   x, z

Proof of Theorem axc9lem2OLD

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axc9lem1 2103	. . . 4 |- ( -. x = y -> ( w = y -> A. x w = y ) )
2		equequ2 1878	. . . . . . 7 |- ( w = y -> ( z = w <-> z = y ) )
3	2	biimprcd 233	. . . . . 6 |- ( z = y -> ( w = y -> z = w ) )
4	3	eximi 1717	. . . . 5 |- ( E. x z = y -> E. x ( w = y -> z = w ) )
5		19.36v 1830	. . . . 5 |- ( E. x ( w = y -> z = w ) <-> ( A. x w = y -> z = w ) )
6	4, 5	sylib 201	. . . 4 |- ( E. x z = y -> ( A. x w = y -> z = w ) )
7	1, 6	syl9 73	. . 3 |- ( -. x = y -> ( E. x z = y -> ( w = y -> z = w ) ) )
8	7	alrimdv 1785	. 2 |- ( -. x = y -> ( E. x z = y -> A. w ( w = y -> z = w ) ) )
9		nfv 1771	. . 3 |- F/ w z = y
10	9, 2	equsal 2138	. 2 |- ( A. w ( w = y -> z = w ) <-> z = y )
11	8, 10	syl6ib 234	1 |- ( -. x = y -> ( E. x z = y -> z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1452   E.wex 1673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678

This theorem is referenced by: (None)