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Theorem ax10lem3 1980
Description: Lemma for ax10 1982. Change free and bound variables. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)
Assertion
Ref Expression
ax10lem3  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Distinct variable group:    z, w

Proof of Theorem ax10lem3
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax10lem1 1978 . . 3  |-  ( A. z  z  =  w  ->  A. v  v  =  w )
2 ax10lem2 1979 . . 3  |-  ( A. v  v  =  w  ->  A. u  u  =  v )
31, 2syl 16 . 2  |-  ( A. z  z  =  w  ->  A. u  u  =  v )
4 ax10lem1 1978 . 2  |-  ( A. u  u  =  v  ->  A. x  x  =  v )
5 ax10lem2 1979 . 2  |-  ( A. x  x  =  v  ->  A. y  y  =  x )
63, 4, 53syl 19 1  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546
This theorem is referenced by:  ax10  1982  a16g  1992  a16gOLD  1993  aev  2025
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 1661  ax-8 1682  ax-6 1736  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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