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Theorem r19.32vdc 2516
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where  ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
Assertion
Ref Expression
r19.32vdc  |-  (DECID  ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  (
ph  \/  A. x  e.  A  ps )
) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.32vdc
StepHypRef Expression
1 r19.21v 2450 . . 3  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
21a1i 9 . 2  |-  (DECID  ph  ->  ( A. x  e.  A  ( -.  ph  ->  ps ) 
<->  ( -.  ph  ->  A. x  e.  A  ps ) ) )
3 dfordc 829 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( -.  ph  ->  ps ) ) )
43ralbidv 2380 . 2  |-  (DECID  ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph  ->  ps )
) )
5 dfordc 829 . 2  |-  (DECID  ph  ->  ( ( ph  \/  A. x  e.  A  ps ) 
<->  ( -.  ph  ->  A. x  e.  A  ps ) ) )
62, 4, 53bitr4d 218 1  |-  (DECID  ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  (
ph  \/  A. x  e.  A  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 664  DECID wdc 780   A.wral 2359
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-dc 781  df-nf 1395  df-ral 2364
This theorem is referenced by: (None)
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