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Theorem xfree2 23041
Description: A partial converse to 19.9t 1794. (Contributed by Stefan Allan, 21-Dec-2008.)
Assertion
Ref Expression
xfree2  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( -.  ph  ->  A. x  -.  ph )
)

Proof of Theorem xfree2
StepHypRef Expression
1 xfree 23040 . 2  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( E. x ph  ->  ph ) )
2 df-ex 1532 . . . . 5  |-  ( E. x ph  <->  -.  A. x  -.  ph )
32imbi1i 315 . . . 4  |-  ( ( E. x ph  ->  ph )  <->  ( -.  A. x  -.  ph  ->  ph )
)
4 con1b 323 . . . 4  |-  ( ( -.  A. x  -.  ph 
->  ph )  <->  ( -.  ph 
->  A. x  -.  ph ) )
53, 4bitri 240 . . 3  |-  ( ( E. x ph  ->  ph )  <->  ( -.  ph  ->  A. x  -.  ph ) )
65albii 1556 . 2  |-  ( A. x ( E. x ph  ->  ph )  <->  A. x
( -.  ph  ->  A. x  -.  ph )
)
71, 6bitri 240 1  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( -.  ph  ->  A. x  -.  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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