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Theorem ax10lem1 1876
Description: Lemma for ax10 1884. Change bound variable. (Contributed by NM, 22-Jul-2015.)
Assertion
Ref Expression
ax10lem1  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Distinct variable groups:    x, w    y, w

Proof of Theorem ax10lem1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 ax-8 1643 . . 3  |-  ( x  =  v  ->  (
x  =  w  -> 
v  =  w ) )
21cbvalivw 1642 . 2  |-  ( A. x  x  =  w  ->  A. v  v  =  w )
3 ax-8 1643 . . 3  |-  ( v  =  y  ->  (
v  =  w  -> 
y  =  w ) )
43cbvalivw 1642 . 2  |-  ( A. v  v  =  w  ->  A. y  y  =  w )
52, 4syl 15 1  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  ax10lem3  1878  ax10lem4  1881  ax10lem5  1882  ax10lem18ALT  29124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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