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Theorem r19.32v 2974
Description: Restricted quantifier version of 19.32v 1778. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
r19.32v  |-  ( 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 r19.32v
StepHypRef Expression
1 r19.21v 2830 . 2  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
2 df-or 371 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32ralbii 2856 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph 
->  ps ) )
4 df-or 371 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
51, 3, 43bitr4i 280 1  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369   A.wral 2775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748
This theorem depends on definitions:  df-bi 188  df-or 371  df-ex 1660  df-ral 2780
This theorem is referenced by:  iinun2  4362  iinuni  4383  axcontlem2  24982  axcontlem7  24987  disjnf  28171  lindslinindsimp2  39530
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