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Theorem r19.32v 2686
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 2630 . 2  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
2 df-or 359 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32ralbii 2567 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph 
->  ps ) )
4 df-or 359 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
51, 3, 43bitr4i 268 1  |-  ( 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 176    \/ wo 357   A.wral 2543
This theorem is referenced by:  iinun2  3968  iinuni  3985  axcontlem2  24593  axcontlem7  24598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-nf 1532  df-ral 2548
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