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Theorem xfree 23904
Description: A partial converse to 19.9t 1789. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
xfree  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( E. x ph  ->  ph ) )

Proof of Theorem xfree
StepHypRef Expression
1 df-nf 1551 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 nf3 1886 . 2  |-  ( F/ x ph  <->  A. x
( E. x ph  ->  ph ) )
31, 2bitr3i 243 1  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( E. x ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem is referenced by:  xfree2  23905
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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