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Theorem 2exbi 26910
Description: Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2exbi  |-  ( A. x A. y ( ph  <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps ) )

Proof of Theorem 2exbi
StepHypRef Expression
1 exbi 1579 . . 3  |-  ( A. y ( ph  <->  ps )  ->  ( E. y ph  <->  E. y ps ) )
21alimi 1546 . 2  |-  ( A. x A. y ( ph  <->  ps )  ->  A. x
( E. y ph  <->  E. y ps ) )
3 exbi 1579 . 2  |-  ( A. x ( E. y ph 
<->  E. y ps )  ->  ( E. x E. y ph  <->  E. x E. y ps ) )
42, 3syl 17 1  |-  ( A. x A. y ( ph  <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   E.wex 1537
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179  df-ex 1538
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