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Theorem r19.32vdc 2580
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 2509 . . 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 877 . . 3  |-  (DECID  ph  ->  ( ( ph  \/  ps ) 
<->  ( -.  ph  ->  ps ) ) )
43ralbidv 2437 . 2  |-  (DECID  ph  ->  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph  ->  ps )
) )
5 dfordc 877 . 2  |-  (DECID  ph  ->  ( ( ph  \/  A. x  e.  A  ps ) 
<->  ( -.  ph  ->  A. x  e.  A  ps ) ) )
62, 4, 53bitr4d 219 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 104    \/ wo 697  DECID wdc 819   A.wral 2416
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-ral 2421
This theorem is referenced by: (None)
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