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Metamath Proof Explorer

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 )