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Metamath Proof Explorer

Theorem nfeqf

Description: A variable is effectively not free in an equality if it is not either of the involved variables. F/ version of ax-c9 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 6-Oct-2016) Remove dependency on ax-11 . (Revised by Wolf Lammen, 6-Sep-2018) (New usage is discouraged.)

Ref Expression
Assertion nfeqf 
|- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )

Proof

Step Hyp Ref Expression
1 nfna1 
 |-  F/ z -. A. z z = x
2 nfna1 
 |-  F/ z -. A. z z = y
3 1 2 nfan 
 |-  F/ z ( -. A. z z = x /\ -. A. z z = y )
4 equvinva 
 |-  ( x = y -> E. w ( x = w /\ y = w ) )
5 dveeq1 
 |-  ( -. A. z z = x -> ( x = w -> A. z x = w ) )
6 5 imp 
 |-  ( ( -. A. z z = x /\ x = w ) -> A. z x = w )
7 dveeq1 
 |-  ( -. A. z z = y -> ( y = w -> A. z y = w ) )
8 7 imp 
 |-  ( ( -. A. z z = y /\ y = w ) -> A. z y = w )
9 equtr2 
 |-  ( ( x = w /\ y = w ) -> x = y )
10 9 alanimi 
 |-  ( ( A. z x = w /\ A. z y = w ) -> A. z x = y )
11 6 8 10 syl2an 
 |-  ( ( ( -. A. z z = x /\ x = w ) /\ ( -. A. z z = y /\ y = w ) ) -> A. z x = y )
12 11 an4s 
 |-  ( ( ( -. A. z z = x /\ -. A. z z = y ) /\ ( x = w /\ y = w ) ) -> A. z x = y )
13 12 ex 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> ( ( x = w /\ y = w ) -> A. z x = y ) )
14 13 exlimdv 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. w ( x = w /\ y = w ) -> A. z x = y ) )
15 4 14 syl5 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
16 3 15 nf5d 
 |-  ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )