Theorem nfeqf 2141

Description: A variable is effectively not free in an equality if it is not either of the involved variables. F/ version of ax-c9 32474. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 1922. (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 1987	. . 3 |- F/ z -. A. z z = x
2		nfna1 1987	. . 3 |- F/ z -. A. z z = y
3	1, 2	nfan 2013	. 2 |- F/ z ( -. A. z z = x /\ -. A. z z = y )
4		equviniv 1874	. . 3 |- ( x = y -> E. w ( x = w /\ y = w ) )
5		dveeq1 2140	. . . . . . . 8 |- ( -. A. z z = x -> ( x = w -> A. z x = w ) )
6	5	imp 431	. . . . . . 7 |- ( ( -. A. z z = x /\ x = w ) -> A. z x = w )
7		dveeq1 2140	. . . . . . . 8 |- ( -. A. z z = y -> ( y = w -> A. z y = w ) )
8	7	imp 431	. . . . . . 7 |- ( ( -. A. z z = y /\ y = w ) -> A. z y = w )
9		equtr2 1871	. . . . . . . 8 |- ( ( x = w /\ y = w ) -> x = y )
10	9	alanimi 1690	. . . . . . 7 |- ( ( A. z x = w /\ A. z y = w ) -> A. z x = y )
11	6, 8, 10	syl2an 480	. . . . . 6 |- ( ( ( -. A. z z = x /\ x = w ) /\ ( -. A. z z = y /\ y = w ) ) -> A. z x = y )
12	11	an4s 836	. . . . 5 |- ( ( ( -. A. z z = x /\ -. A. z z = y ) /\ ( x = w /\ y = w ) ) -> A. z x = y )
13	12	ex 436	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( ( x = w /\ y = w ) -> A. z x = y ) )
14	13	exlimdv 1781	. . 3 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. w ( x = w /\ y = w ) -> A. z x = y ) )
15	4, 14	syl5 33	. 2 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( x = y -> A. z x = y ) )
16	3, 15	nfd 1958	1 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   A.wal 1444   E.wex 1665   F/wnf 1669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-12 1935  ax-13 2093

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670

This theorem is referenced by:  axc9  2142  dvelimf  2170  equveli  2182  2ax6elem  2280  wl-exeq  31879  wl-nfeqfb  31882  wl-equsb4  31897  wl-2sb6d  31900  wl-sbalnae  31904