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Theorem 19.37vv 27574
Description: Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.37vv  |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
Distinct variable groups:    ps, x    ps, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem 19.37vv
StepHypRef Expression
1 19.37v 1923 . . 3  |-  ( E. y ( ps  ->  ph )  <->  ( ps  ->  E. y ph ) )
21exbii 1593 . 2  |-  ( E. x E. y ( ps  ->  ph )  <->  E. x
( ps  ->  E. y ph ) )
3 19.37v 1923 . 2  |-  ( E. x ( ps  ->  E. y ph )  <->  ( ps  ->  E. x E. y ph ) )
42, 3bitri 242 1  |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   E.wex 1551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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