Theorem aeveq 1982

Description: The antecedent A. x x = y with a dv condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)

Assertion

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

Distinct variable group:   x, y

Proof of Theorem aeveq

Dummy variable u is distinct from all other variables.

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

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

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

This theorem is referenced by:  aev  1983  2ax6e  2450  aevdemo  27317  wl-spae  33306