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Theorem e2ebind 28628
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 28628 is derived from e2ebindVD 29004. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
e2ebind  |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
)

Proof of Theorem e2ebind
StepHypRef Expression
1 nfe1 1718 . . . 4  |-  F/ y E. y ph
2119.9 1795 . . 3  |-  ( E. y E. y ph  <->  E. y ph )
3 biidd 228 . . . . . 6  |-  ( A. y  y  =  x  ->  ( ph  <->  ph ) )
43drex1 1920 . . . . 5  |-  ( A. y  y  =  x  ->  ( E. y ph  <->  E. x ph ) )
54drex2 1921 . . . 4  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. y E. x ph ) )
6 excom 1798 . . . 4  |-  ( E. y E. x ph  <->  E. x E. y ph )
75, 6syl6bb 252 . . 3  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. x E. y ph ) )
82, 7syl5rbbr 251 . 2  |-  ( A. y  y  =  x  ->  ( E. x E. y ph  <->  E. y ph )
)
98aecoms 1900 1  |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531    = wceq 1632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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