MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.32v Unicode version

Theorem r19.32v 2687
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 2631 . 2  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
2 df-or 361 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32ralbii 2568 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph 
->  ps ) )
4 df-or 361 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
51, 3, 43bitr4i 270 1  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359   A.wral 2544
This theorem is referenced by:  iinun2  3969  iinuni  3986  axcontlem2  24000  axcontlem7  24005
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-11 1719
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-nf 1537  df-ral 2549
  Copyright terms: Public domain W3C validator