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Theorem r19.32v 2861
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (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 2798 . 2  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
2 df-or 370 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32ralbii 2734 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph 
->  ps ) )
4 df-or 370 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
51, 3, 43bitr4i 277 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 184    \/ wo 368   A.wral 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-ral 2715
This theorem is referenced by:  iinun2  4231  iinuni  4249  axcontlem2  23162  axcontlem7  23167  disjnf  25867  lindslinindsimp2  30886
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