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Theorem nfeqf 2100
Description: A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-c9 32381. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 1892. (Revised by Wolf Lammen, 6-Sep-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 nfna1 1958 . . 3  |-  F/ z  -.  A. z  z  =  x
2 nfna1 1958 . . 3  |-  F/ z  -.  A. z  z  =  y
31, 2nfan 1984 . 2  |-  F/ z ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )
4 equviniv 1853 . . 3  |-  ( x  =  y  ->  E. w
( x  =  w  /\  y  =  w ) )
5 dveeq1 2099 . . . . . . . 8  |-  ( -. 
A. z  z  =  x  ->  ( x  =  w  ->  A. z  x  =  w )
)
65imp 430 . . . . . . 7  |-  ( ( -.  A. z  z  =  x  /\  x  =  w )  ->  A. z  x  =  w )
7 dveeq1 2099 . . . . . . . 8  |-  ( -. 
A. z  z  =  y  ->  ( y  =  w  ->  A. z 
y  =  w ) )
87imp 430 . . . . . . 7  |-  ( ( -.  A. z  z  =  y  /\  y  =  w )  ->  A. z 
y  =  w )
9 equtr2 1852 . . . . . . . 8  |-  ( ( x  =  w  /\  y  =  w )  ->  x  =  y )
109alanimi 1684 . . . . . . 7  |-  ( ( A. z  x  =  w  /\  A. z 
y  =  w )  ->  A. z  x  =  y )
116, 8, 10syl2an 479 . . . . . 6  |-  ( ( ( -.  A. z 
z  =  x  /\  x  =  w )  /\  ( -.  A. z 
z  =  y  /\  y  =  w )
)  ->  A. z  x  =  y )
1211an4s 833 . . . . 5  |-  ( ( ( -.  A. z 
z  =  x  /\  -.  A. z  z  =  y )  /\  (
x  =  w  /\  y  =  w )
)  ->  A. z  x  =  y )
1312ex 435 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( (
x  =  w  /\  y  =  w )  ->  A. z  x  =  y ) )
1413exlimdv 1768 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( E. w ( x  =  w  /\  y  =  w )  ->  A. z  x  =  y )
)
154, 14syl5 33 . 2  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  A. z  x  =  y )
)
163, 15nfd 1929 1  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1659   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  axc9  2101  dvelimf  2131  equveli  2143  2ax6elem  2244  wl-exeq  31781  wl-nfeqfb  31784  wl-equsb4  31799  wl-2sb6d  31802  wl-sbalnae  31806
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