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Theorem axc9lem2 2143
Description: Lemma for nfeqf2 2145. This lemma is equivalent to ax13v 2102 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axc9lem2  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
Distinct variable group:    x, z

Proof of Theorem axc9lem2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 axc9lem1 2103 . . . 4  |-  ( -.  x  =  y  -> 
( w  =  y  ->  A. x  w  =  y ) )
2 equequ2 1878 . . . . . . 7  |-  ( w  =  y  ->  (
z  =  w  <->  z  =  y ) )
32biimprcd 233 . . . . . 6  |-  ( z  =  y  ->  (
w  =  y  -> 
z  =  w ) )
43eximi 1717 . . . . 5  |-  ( E. x  z  =  y  ->  E. x ( w  =  y  ->  z  =  w ) )
5 19.36v 1830 . . . . 5  |-  ( E. x ( w  =  y  ->  z  =  w )  <->  ( A. x  w  =  y  ->  z  =  w ) )
64, 5sylib 201 . . . 4  |-  ( E. x  z  =  y  ->  ( A. x  w  =  y  ->  z  =  w ) )
71, 6syl9 73 . . 3  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  (
w  =  y  -> 
z  =  w ) ) )
87alrimdv 1785 . 2  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  A. w
( w  =  y  ->  z  =  w ) ) )
92equsalvw 1857 . 2  |-  ( A. w ( w  =  y  ->  z  =  w )  <->  z  =  y )
108, 9syl6ib 234 1  |-  ( -.  x  =  y  -> 
( E. x  z  =  y  ->  z  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1452   E.wex 1673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  nfeqf2  2145
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