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Theorem xfree 23978
Description: A partial converse to 19.9t 1795. (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 1555 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 nf3 1892 . 2  |-  ( F/ x ph  <->  A. x
( E. x ph  ->  ph ) )
31, 2bitr3i 244 1  |-  ( A. x ( ph  ->  A. x ph )  <->  A. x
( E. x ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551   F/wnf 1554
This theorem is referenced by:  xfree2  23979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-11 1763
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-nf 1555
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