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Theorem 19.36vv 26993
Description: Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
19.36vv  |-  ( E. x E. y (
ph  ->  ps )  <->  ( A. x A. y ph  ->  ps ) )
Distinct variable groups:    ps, x    ps, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem 19.36vv
StepHypRef Expression
1 19.36v 1837 . . 3  |-  ( E. y ( ph  ->  ps )  <->  ( A. y ph  ->  ps ) )
21exbii 1569 . 2  |-  ( E. x E. y (
ph  ->  ps )  <->  E. x
( A. y ph  ->  ps ) )
3 19.36v 1837 . 2  |-  ( E. x ( A. y ph  ->  ps )  <->  ( A. x A. y ph  ->  ps ) )
42, 3bitri 240 1  |-  ( E. x E. y (
ph  ->  ps )  <->  ( A. x A. y ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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