Theorem nfeqf 2045

Description: A variable is effectively not free in an equality if it is not either of the involved variables. F/ version of ax-c9 2221. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 1842. (Revised by Wolf Lammen, 6-Sep-2018.)

Assertion

Ref	Expression
nfeqf

Proof of Theorem nfeqf

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfna1 1903	. . 3
2		nfna1 1903	. . 3
3	1, 2	nfan 1928	. 2
4		equviniv 1803	. . 3
5		dveeq1 2044	. . . . . . . 8
6	5	imp 429	. . . . . . 7
7		dveeq1 2044	. . . . . . . 8
8	7	imp 429	. . . . . . 7
9		equtr2 1802	. . . . . . . 8
10	9	alanimi 1637	. . . . . . 7
11	6, 8, 10	syl2an 477	. . . . . 6
12	11	an4s 826	. . . . 5
13	12	ex 434	. . . 4
14	13	exlimdv 1724	. . 3
15	4, 14	syl5 32	. 2
16	3, 15	nfd 1878	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612  F/wnf 1616

This theorem is referenced by:  axc9  2046  dvelimf  2076  equveli  2088  equveliOLD  2089  2ax6elem  2193  wl-exeq  29987  wl-nfeqfb  29990  wl-equsb4  30005  wl-2sb6d  30008  wl-sbalnae  30012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617