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Theorem 19.32dc 1610
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 1589 . . . 4  |-  F/ x  -.  ph
3219.21 1516 . . 3  |-  ( A. x ( -.  ph  ->  ps )  <->  ( -.  ph 
->  A. x ps )
)
43a1i 9 . 2  |-  (DECID  ph  ->  ( A. x ( -. 
ph  ->  ps )  <->  ( -.  ph 
->  A. x ps )
) )
51nfdc 1590 . . 3  |-  F/ xDECID  ph
6 dfordc 825 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( -.  ph  ->  ps ) ) )
75, 6albid 1547 . 2  |-  (DECID  ph  ->  ( A. x ( ph  \/  ps )  <->  A. x
( -.  ph  ->  ps ) ) )
8 dfordc 825 . 2  |-  (DECID  ph  ->  ( ( ph  \/  A. x ps )  <->  ( -.  ph 
->  A. x ps )
) )
94, 7, 83bitr4d 218 1  |-  (DECID  ph  ->  ( A. x ( ph  \/  ps )  <->  ( ph  \/  A. x ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662  DECID wdc 776   A.wal 1283   F/wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-gen 1379  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by: (None)
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