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 )