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Theorem 19.37vv 27253
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 1911 . . 3  |-  ( E. y ( ps  ->  ph )  <->  ( ps  ->  E. y ph ) )
21exbii 1589 . 2  |-  ( E. x E. y ( ps  ->  ph )  <->  E. x
( ps  ->  E. y ph ) )
3 19.37v 1911 . 2  |-  ( E. x ( ps  ->  E. y ph )  <->  ( ps  ->  E. x E. y ph ) )
42, 3bitri 241 1  |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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