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Theorem e2ebind 31569
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 31569 is derived from e2ebindVD 31945. (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 1780 . . . 4  |-  F/ y E. y ph
2119.9 1829 . . 3  |-  ( E. y E. y ph  <->  E. y ph )
3 biidd 237 . . . . . 6  |-  ( A. y  y  =  x  ->  ( ph  <->  ph ) )
43drex1 2026 . . . . 5  |-  ( A. y  y  =  x  ->  ( E. y ph  <->  E. x ph ) )
54drex2 2027 . . . 4  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. y E. x ph ) )
6 excom 1789 . . . 4  |-  ( E. y E. x ph  <->  E. x E. y ph )
75, 6syl6bb 261 . . 3  |-  ( A. y  y  =  x  ->  ( E. y E. y ph  <->  E. x E. y ph ) )
82, 7syl5rbbr 260 . 2  |-  ( A. y  y  =  x  ->  ( E. x E. y ph  <->  E. y ph )
)
98aecoms 2009 1  |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by: (None)
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