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Metamath Proof Explorer

Theorem hbae-o

Description: All variables are effectively bound in an identical variable specifier. Version of hbae using ax-c11 . (Contributed by NM, 13-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 ax-c5 
 |-  ( A. x x = y -> x = y )
2 ax-c9 
 |-  ( -. 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 ax-c11 
 |-  ( A. x x = z -> ( A. x x = y -> A. z x = y ) )
5 4 aecoms-o 
 |-  ( A. z z = x -> ( A. x x = y -> A. z x = y ) )
6 ax-c11 
 |-  ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
7 6 pm2.43i 
 |-  ( A. x x = y -> A. y x = y )
8 ax-c11 
 |-  ( A. y y = z -> ( A. y x = y -> A. z x = y ) )
9 7 8 syl5 
 |-  ( A. y y = z -> ( A. x x = y -> A. z x = y ) )
10 9 aecoms-o 
 |-  ( A. z z = y -> ( A. x x = y -> A. z x = y ) )
11 3 5 10 pm2.61ii 
 |-  ( A. x x = y -> A. z x = y )
12 11 axc4i-o 
 |-  ( A. x x = y -> A. x A. z x = y )
13 ax-11 
 |-  ( A. x A. z x = y -> A. z A. x x = y )
14 12 13 syl 
 |-  ( A. x x = y -> A. z A. x x = y )