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Theorem nfeqf 1930
Description: A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2114. (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 1928 . . 3  |-  F/ z  -.  A. z  z  =  x
2 nfnae 1928 . . 3  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 1800 . 2  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 ax12o 1906 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y )
) )
54imp 418 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
63, 5nfd 1770 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 358   A.wal 1531   F/wnf 1535
This theorem is referenced by:  equvini  1959  equveli  1960  nfsb4t  2052  sbcom  2061  nfeud2  2188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536
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