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Theorem nfeqf 2141
 Description: A variable is effectively not free in an equality if it is not either of the involved variables. 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

Proof of Theorem nfeqf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfna1 1987 . . 3
2 nfna1 1987 . . 3
31, 2nfan 2013 . 2
4 equviniv 1874 . . 3
5 dveeq1 2140 . . . . . . . 8
65imp 431 . . . . . . 7
7 dveeq1 2140 . . . . . . . 8
87imp 431 . . . . . . 7
9 equtr2 1871 . . . . . . . 8
109alanimi 1690 . . . . . . 7
116, 8, 10syl2an 480 . . . . . 6
1211an4s 836 . . . . 5
1312ex 436 . . . 4
1413exlimdv 1781 . . 3
154, 14syl5 33 . 2
163, 15nfd 1958 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1444  wex 1665  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
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