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Theorem cbvaev 1896
Description: Change bound variable in an equality with a dv condition. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
Assertion
Ref Expression
cbvaev  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Distinct variable groups:    x, w    y, w

Proof of Theorem cbvaev
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 ax7 1870 . . 3  |-  ( x  =  v  ->  (
x  =  w  -> 
v  =  w ) )
21cbvalivw 1860 . 2  |-  ( A. x  x  =  w  ->  A. v  v  =  w )
3 ax7 1870 . . 3  |-  ( v  =  y  ->  (
v  =  w  -> 
y  =  w ) )
43cbvalivw 1860 . 2  |-  ( A. v  v  =  w  ->  A. y  y  =  w )
52, 4syl 17 1  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452
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
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  axc11nlem  2031  aevlem1  2032  axc11nlemALT  2152  aevlem1ALT  2164  bj-axc11nlemv  31391  bj-aevlem1v  31396
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