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Theorem aev 2048
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 2041 . 2  |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
2 aevlem1 2047 . . 3  |-  ( A. x  x  =  y  ->  A. u  u  =  v )
3 ax-8 1688 . . . 4  |-  ( u  =  w  ->  (
u  =  v  ->  w  =  v )
)
43spimv 1964 . . 3  |-  ( A. u  u  =  v  ->  w  =  v )
52, 4syl 16 . 2  |-  ( A. x  x  =  y  ->  w  =  v )
61, 5alrimih 1575 1  |-  ( A. x  x  =  y  ->  A. z  w  =  v )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem is referenced by:  ax16ALT2  2157  a16gALT  2158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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