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Theorem axc9lem1 2106
Description: A version of ax13v 2105 with one distinct variable restriction dropped. For convenience,  y is kept on the right side of equations. The proof of ax13 2155 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
axc9lem1  |-  ( -.  x  =  y  -> 
( z  =  y  ->  A. x  z  =  y ) )
Distinct variable group:    x, z

Proof of Theorem axc9lem1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 equviniv 1880 . 2  |-  ( z  =  y  ->  E. w
( z  =  w  /\  y  =  w ) )
2 ax13v 2105 . . . . 5  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
3 equequ2 1876 . . . . . . 7  |-  ( y  =  w  ->  (
z  =  y  <->  z  =  w ) )
43biimprcd 233 . . . . . 6  |-  ( z  =  w  ->  (
y  =  w  -> 
z  =  y ) )
54alimdv 1771 . . . . 5  |-  ( z  =  w  ->  ( A. x  y  =  w  ->  A. x  z  =  y ) )
62, 5syl9 72 . . . 4  |-  ( -.  x  =  y  -> 
( z  =  w  ->  ( y  =  w  ->  A. x  z  =  y )
) )
76impd 438 . . 3  |-  ( -.  x  =  y  -> 
( ( z  =  w  /\  y  =  w )  ->  A. x  z  =  y )
)
87exlimdv 1787 . 2  |-  ( -.  x  =  y  -> 
( E. w ( z  =  w  /\  y  =  w )  ->  A. x  z  =  y ) )
91, 8syl5 32 1  |-  ( -.  x  =  y  -> 
( z  =  y  ->  A. x  z  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376   A.wal 1450   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672
This theorem is referenced by:  ax6e  2107  axc9lem2  2146  axc9lem2OLD  2147  nfeqf2  2148
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