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Theorem 19.32dc 1659
Description: Theorem 19.32 of [Margaris] p. 90, where  ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
Hypothesis
Ref Expression
19.32dc.1  |-  F/ x ph
Assertion
Ref Expression
19.32dc  |-  (DECID  ph  ->  ( A. x ( ph  \/  ps )  <->  ( ph  \/  A. x ps )
) )

Proof of Theorem 19.32dc
StepHypRef Expression
1 19.32dc.1 . . . . 5  |-  F/ x ph
21nfn 1638 . . . 4  |-  F/ x  -.  ph
3219.21 1563 . . 3  |-  ( A. x ( -.  ph  ->  ps )  <->  ( -.  ph 
->  A. x ps )
)
43a1i 9 . 2  |-  (DECID  ph  ->  ( A. x ( -. 
ph  ->  ps )  <->  ( -.  ph 
->  A. x ps )
) )
51nfdc 1639 . . 3  |-  F/ xDECID  ph
6 dfordc 878 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( -.  ph  ->  ps ) ) )
75, 6albid 1595 . 2  |-  (DECID  ph  ->  ( A. x ( ph  \/  ps )  <->  A. x
( -.  ph  ->  ps ) ) )
8 dfordc 878 . 2  |-  (DECID  ph  ->  ( ( ph  \/  A. x ps )  <->  ( -.  ph 
->  A. x ps )
) )
94, 7, 83bitr4d 219 1  |-  (DECID  ph  ->  ( A. x ( ph  \/  ps )  <->  ( ph  \/  A. x ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 698  DECID wdc 820   A.wal 1333   F/wnf 1440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-gen 1429  ax-ie2 1474  ax-4 1490  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441
This theorem is referenced by: (None)
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