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Theorem 2exanali 27689
Description: Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2exanali  |-  ( -. 
E. x E. y
( ph  /\  -.  ps ) 
<-> 
A. x A. y
( ph  ->  ps )
)

Proof of Theorem 2exanali
StepHypRef Expression
1 2nalexn 1563 . . 3  |-  ( -. 
A. x A. y
( ph  ->  ps )  <->  E. x E. y  -.  ( ph  ->  ps ) )
21con1bii 321 . 2  |-  ( -. 
E. x E. y  -.  ( ph  ->  ps ) 
<-> 
A. x A. y
( ph  ->  ps )
)
3 annim 414 . . 3  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
432exbii 1573 . 2  |-  ( E. x E. y (
ph  /\  -.  ps )  <->  E. x E. y  -.  ( ph  ->  ps ) )
52, 4xchnxbir 300 1  |-  ( -. 
E. x E. y
( ph  /\  -.  ps ) 
<-> 
A. x A. y
( ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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