Theorem hbae

Description: All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbaev when possible. (Contributed by NM, 13-May-1993) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sp	|- ( A. x x = y -> x = y )
2		axc9	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
3	1 2	syl7	|- ( -. A. z z = x -> ( -. A. z z = y -> ( A. x x = y -> A. z x = y ) ) )
4		axc11r	|- ( A. z z = x -> ( A. x x = y -> A. z x = y ) )
5		axc11	|- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
6	5	pm2.43i	|- ( A. x x = y -> A. y x = y )
7		axc11r	|- ( A. z z = y -> ( A. y x = y -> A. z x = y ) )
8	6 7	syl5	|- ( A. z z = y -> ( A. x x = y -> A. z x = y ) )
9	3 4 8	pm2.61ii	|- ( A. x x = y -> A. z x = y )
10	9	axc4i	|- ( A. x x = y -> A. x A. z x = y )
11		ax-11	|- ( A. x A. z x = y -> A. z A. x x = y )
12	10 11	syl	|- ( A. x x = y -> A. z A. x x = y )