Theorem nfeqf 2301

Description: A variable is effectively not free in an equality if it is not either of the involved variables. F/ version of ax-c9 34175. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 6-Sep-2018.)

Assertion

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

Proof of Theorem nfeqf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfna1 2029	. . 3 |- F/ z -. A. z z = x
2		nfna1 2029	. . 3 |- F/ z -. A. z z = y
3	1, 2	nfan 1828	. 2 |- F/ z ( -. A. z z = x /\ -. A. z z = y )
4		equviniva 1960	. . 3 |- ( x = y -> E. w ( x = w /\ y = w ) )
5		dveeq1 2300	. . . . . . . 8 |- ( -. A. z z = x -> ( x = w -> A. z x = w ) )
6	5	imp 445	. . . . . . 7 |- ( ( -. A. z z = x /\ x = w ) -> A. z x = w )
7		dveeq1 2300	. . . . . . . 8 |- ( -. A. z z = y -> ( y = w -> A. z y = w ) )
8	7	imp 445	. . . . . . 7 |- ( ( -. A. z z = y /\ y = w ) -> A. z y = w )
9		equtr2 1954	. . . . . . . 8 |- ( ( x = w /\ y = w ) -> x = y )
10	9	alanimi 1744	. . . . . . 7 |- ( ( A. z x = w /\ A. z y = w ) -> A. z x = y )
11	6, 8, 10	syl2an 494	. . . . . 6 |- ( ( ( -. A. z z = x /\ x = w ) /\ ( -. A. z z = y /\ y = w ) ) -> A. z x = y )
12	11	an4s 869	. . . . 5 |- ( ( ( -. A. z z = x /\ -. A. z z = y ) /\ ( x = w /\ y = w ) ) -> A. z x = y )
13	12	ex 450	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( ( x = w /\ y = w ) -> A. z x = y ) )
14	13	exlimdv 1861	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. w ( x = w /\ y = w ) -> A. z x = y ) )
15	4, 14	syl5 34	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
16	3, 15	nf5d 2118	1 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  axc9  2302  dvelimf  2334  equvel  2347  2ax6elem  2449  wl-exeq  33321  wl-nfeqfb  33323  wl-equsb4  33338  wl-2sb6d  33341  wl-sbalnae  33345