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Theorem nfeqf 2010
Description: A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2220. (Contributed by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Apr-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.
StepHypRef Expression
1 ax12v 1952 . . 3  |-  ( -.  z  =  x  -> 
( x  =  w  ->  A. z  x  =  w ) )
21ax12olem3 2008 . 2  |-  ( -. 
A. z  z  =  x  ->  F/ z  x  =  w )
3 ax12v 1952 . . 3  |-  ( -.  z  =  y  -> 
( y  =  w  ->  A. z  y  =  w ) )
43ax12olem3 2008 . 2  |-  ( -. 
A. z  z  =  y  ->  F/ z 
y  =  w )
52, 4ax12olem4 2009 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   F/wnf 1554
This theorem is referenced by:  ax12o  2011  dvelimf  2069  equvini  2084  equviniOLD  2085  equveli  2086  equveliOLD  2087  nfsb4tOLD  2129  sbcom  2165  sbcomOLD  2166  nfeud2  2294  wl-exeq  26235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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