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Theorem 19.12vv 1839
Description: Special case of 19.12 1734 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
19.12vv  |-  ( E. x A. y (
ph  ->  ps )  <->  A. y E. x ( ph  ->  ps ) )
Distinct variable groups:    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem 19.12vv
StepHypRef Expression
1 19.21v 1831 . . 3  |-  ( A. y ( ph  ->  ps )  <->  ( ph  ->  A. y ps ) )
21exbii 1569 . 2  |-  ( E. x A. y (
ph  ->  ps )  <->  E. x
( ph  ->  A. y ps ) )
3 nfv 1605 . . . 4  |-  F/ x ps
43nfal 1766 . . 3  |-  F/ x A. y ps
5419.36 1807 . 2  |-  ( E. x ( ph  ->  A. y ps )  <->  ( A. x ph  ->  A. y ps ) )
6 19.36v 1837 . . . 4  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  ps ) )
76albii 1553 . . 3  |-  ( A. y E. x ( ph  ->  ps )  <->  A. y
( A. x ph  ->  ps ) )
8 nfv 1605 . . . . 5  |-  F/ y
ph
98nfal 1766 . . . 4  |-  F/ y A. x ph
10919.21 1791 . . 3  |-  ( A. y ( A. x ph  ->  ps )  <->  ( A. x ph  ->  A. y ps ) )
117, 10bitr2i 241 . 2  |-  ( ( A. x ph  ->  A. y ps )  <->  A. y E. x ( ph  ->  ps ) )
122, 5, 113bitri 262 1  |-  ( E. x A. y (
ph  ->  ps )  <->  A. y E. x ( 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-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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