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Theorem 2exbi 31529
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 1671 . . 3  |-  ( A. y ( ph  <->  ps )  ->  ( E. y ph  <->  E. y ps ) )
21alimi 1638 . 2  |-  ( A. x A. y ( ph  <->  ps )  ->  A. x
( E. y ph  <->  E. y ps ) )
3 exbi 1671 . 2  |-  ( A. x ( E. y ph 
<->  E. y ps )  ->  ( E. x E. y ph  <->  E. x E. y ps ) )
42, 3syl 16 1  |-  ( A. x A. y ( ph  <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-ex 1618
This theorem is referenced by: (None)
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