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Theorem xfree2 23901
Description: A partial converse to 19.9t 1789. (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 23900 . 2  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( E. x ph  ->  ph ) )
2 df-ex 1548 . . . . 5  |-  ( E. x ph  <->  -.  A. x  -.  ph )
32imbi1i 316 . . . 4  |-  ( ( E. x ph  ->  ph )  <->  ( -.  A. x  -.  ph  ->  ph )
)
4 con1b 324 . . . 4  |-  ( ( -.  A. x  -.  ph 
->  ph )  <->  ( -.  ph 
->  A. x  -.  ph ) )
53, 4bitri 241 . . 3  |-  ( ( E. x ph  ->  ph )  <->  ( -.  ph  ->  A. x  -.  ph ) )
65albii 1572 . 2  |-  ( A. x ( E. x ph  ->  ph )  <->  A. x
( -.  ph  ->  A. x  -.  ph )
)
71, 6bitri 241 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 177   A.wal 1546   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 1662  ax-8 1683  ax-6 1740  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-ex 1548  df-nf 1551
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