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Metamath Proof Explorer

Theorem equvel

Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini . Usage of this theorem is discouraged because it depends on ax-13 . Use equvelv when possible. (Contributed by NM, 1-Mar-2013) (Proof shortened by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 albi 
 |-  ( A. z ( z = x <-> z = y ) -> ( A. z z = x <-> A. z z = y ) )
2 ax6e 
 |-  E. z z = y
3 biimpr 
 |-  ( ( z = x <-> z = y ) -> ( z = y -> z = x ) )
4 ax7 
 |-  ( z = x -> ( z = y -> x = y ) )
5 3 4 syli 
 |-  ( ( z = x <-> z = y ) -> ( z = y -> x = y ) )
6 5 com12 
 |-  ( z = y -> ( ( z = x <-> z = y ) -> x = y ) )
7 2 6 eximii 
 |-  E. z ( ( z = x <-> z = y ) -> x = y )
8 7 19.35i 
 |-  ( A. z ( z = x <-> z = y ) -> E. z x = y )
9 4 spsd 
 |-  ( z = x -> ( A. z z = y -> x = y ) )
10 9 sps 
 |-  ( A. z z = x -> ( A. z z = y -> x = y ) )
11 10 a1dd 
 |-  ( A. z z = x -> ( A. z z = y -> ( E. z x = y -> x = y ) ) )
12 nfeqf 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
13 12 19.9d 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. z x = y -> x = y ) )
14 13 ex 
 |-  ( -. A. z z = x -> ( -. A. z z = y -> ( E. z x = y -> x = y ) ) )
15 11 14 bija 
 |-  ( ( A. z z = x <-> A. z z = y ) -> ( E. z x = y -> x = y ) )
16 1 8 15 sylc 
 |-  ( A. z ( z = x <-> z = y ) -> x = y )