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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.
StepHypRef Expression
1 nfna1 1903 . . 3
2 nfna1 1903 . . 3
31, 2nfan 1928 . 2
4 equviniv 1803 . . 3
5 dveeq1 2044 . . . . . . . 8
65imp 429 . . . . . . 7
7 dveeq1 2044 . . . . . . . 8
87imp 429 . . . . . . 7
9 equtr2 1802 . . . . . . . 8
109alanimi 1637 . . . . . . 7
116, 8, 10syl2an 477 . . . . . 6
1211an4s 826 . . . . 5
1312ex 434 . . . 4
1413exlimdv 1724 . . 3
154, 14syl5 32 . 2
163, 15nfd 1878 1
 Colors of variables: wff setvar class 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
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