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Theorem ax10lem5 1895
Description: Lemma for ax10 1897. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)
Assertion
Ref Expression
ax10lem5  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Distinct variable group:    z, w

Proof of Theorem ax10lem5
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax10lem1 1889 . . . 4  |-  ( A. z  z  =  w  ->  A. v  v  =  w )
2 ax10lem4 1894 . . . 4  |-  ( A. v  v  =  w  ->  A. u  u  =  v )
31, 2syl 15 . . 3  |-  ( A. z  z  =  w  ->  A. u  u  =  v )
4 ax10lem1 1889 . . 3  |-  ( A. u  u  =  v  ->  A. x  x  =  v )
53, 4syl 15 . 2  |-  ( A. z  z  =  w  ->  A. x  x  =  v )
6 ax10lem4 1894 . 2  |-  ( A. x  x  =  v  ->  A. y  y  =  x )
75, 6syl 15 1  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530
This theorem is referenced by:  ax10  1897  a16g  1898  aev  1944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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