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Theorem aev 1931
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)
Assertion
Ref Expression
aev  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Distinct variable group:    x, y

Proof of Theorem aev
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 hbae 1893 . 2  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
2 ax10lem5 1882 . . 3  |-  ( A. x  x  =  y  ->  A. u  u  =  v )
3 ax-8 1643 . . . 4  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
43spimv 1930 . . 3  |-  ( A. u  u  =  v  ->  w  =  v )
52, 4syl 15 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
61, 5alrimih 1552 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  ax16ALT2  1988  a16gALT  1989
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  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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