Theorem wl-hbnaev 33305

Description: Any variable is free in -. A. x x = y, if x and y are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2019. (Contributed by Wolf Lammen, 9-Apr-2021.)

Assertion

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

Distinct variable group:   x, y

Proof of Theorem wl-hbnaev

Dummy variables u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1983	. . 3 |- ( A. u u = t -> A. x x = y )
2	1	con3i 150	. 2 |- ( -. A. x x = y -> -. A. u u = t )
3		ax-5 1839	. 2 |- ( -. A. u u = t -> A. z -. A. u u = t )
4		aev 1983	. . . 4 |- ( A. x x = y -> A. u u = t )
5	4	con3i 150	. . 3 |- ( -. A. u u = t -> -. A. x x = y )
6	5	alimi 1739	. 2 |- ( A. z -. A. u u = t -> A. z -. A. x x = y )
7	2, 3, 6	3syl 18	1 |- ( -. A. x x = y -> A. z -. A. x x = y )

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

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

This theorem is referenced by: (None)