Theorem aevOLD 2162

Description: Obsolete proof of aev 1983 as of 19-Mar-2021. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2246, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
aevOLD	|- ( A. x x = y -> A. z w = v )

Distinct variable group:   x, y

Proof of Theorem aevOLD

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 1981	. . 3 |- ( A. x x = y -> A. u u = w )
2		ax6ev 1890	. . . 4 |- E. u u = v
3		ax7 1943	. . . . 5 |- ( u = w -> ( u = v -> w = v ) )
4	3	aleximi 1759	. . . 4 |- ( A. u u = w -> ( E. u u = v -> E. u w = v ) )
5	2, 4	mpi 20	. . 3 |- ( A. u u = w -> E. u w = v )
6		ax5e 1841	. . 3 |- ( E. u w = v -> w = v )
7	1, 5, 6	3syl 18	. 2 |- ( A. x x = y -> w = v )
8		axc16g 2134	. 2 |- ( A. x x = y -> ( w = v -> A. z w = v ) )
9	7, 8	mpd 15	1 |- ( A. x x = y -> A. z w = v )

Syntax hints:   -> wi 4   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-12 2047

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

This theorem is referenced by: (None)