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Theorem e2ebind 27994
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 27994 is derived from e2ebindVD 28366. (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 1739 . . . 4  |-  F/ y E. y ph
2119.9 1791 . . 3  |-  ( E. y E. y ph  <->  E. y ph )
3 biidd 229 . . . . . 6  |-  ( A. y  y  =  x  ->  ( ph  <->  ph ) )
43drex1 2008 . . . . 5  |-  ( A. y  y  =  x  ->  ( E. y ph  <->  E. x ph ) )
54drex2 2009 . . . 4  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. y E. x ph ) )
6 excom 1748 . . . 4  |-  ( E. y E. x ph  <->  E. x E. y ph )
75, 6syl6bb 253 . . 3  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. x E. y ph ) )
82, 7syl5rbbr 252 . 2  |-  ( A. y  y  =  x  ->  ( E. x E. y ph  <->  E. y ph )
)
98aecoms 1997 1  |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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