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Theorem ax10lem1 2022
Description: Lemma for ax10 2025. 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 1687 . . 3  |-  ( x  =  v  ->  (
x  =  w  -> 
v  =  w ) )
21cbvalivw 1686 . 2  |-  ( A. x  x  =  w  ->  A. v  v  =  w )
3 ax-8 1687 . . 3  |-  ( v  =  y  ->  (
v  =  w  -> 
y  =  w ) )
43cbvalivw 1686 . 2  |-  ( A. v  v  =  w  ->  A. y  y  =  w )
52, 4syl 16 1  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549
This theorem is referenced by:  ax10lem2  2023  ax10lem3OLD  2027  ax10lem4OLD  2030  ax10lem5OLD  2031  aevlem1  2046  ax10lem4NEW7  29471  ax10lem5NEW7  29472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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