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Theorem nfeqf 2014
Description: A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2192. (Contributed by Mario Carneiro, 6-Oct-2016.)
Assertion
Ref Expression
nfeqf  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )

Proof of Theorem nfeqf
StepHypRef Expression
1 nfnae 2008 . . 3  |-  F/ z  -.  A. z  z  =  x
2 nfnae 2008 . . 3  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 1842 . 2  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 ax12o 1976 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
54imp 419 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
63, 5nfd 1778 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 359   A.wal 1546   F/wnf 1550
This theorem is referenced by:  dvelimh  2015  equvini  2040  equviniOLD  2041  equveli  2042  equveliOLD  2043  nfsb4t  2129  sbcom  2138  nfeud2  2266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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