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Theorem r19.32 27922
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 2855. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Hypothesis
Ref Expression
r19.32.1  |-  F/ x ph
Assertion
Ref Expression
r19.32  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )

Proof of Theorem r19.32
StepHypRef Expression
1 r19.32.1 . . . 4  |-  F/ x ph
21nfn 1812 . . 3  |-  F/ x  -.  ph
32r19.21 2793 . 2  |-  ( A. x  e.  A  ( -.  ph  ->  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
4 df-or 361 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
54ralbii 2730 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ph 
->  ps ) )
6 df-or 361 . 2  |-  ( (
ph  \/  A. x  e.  A  ps )  <->  ( -.  ph  ->  A. x  e.  A  ps )
)
73, 5, 63bitr4i 270 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 178    \/ wo 359   F/wnf 1554   A.wral 2706
This theorem is referenced by:  2reu3  27943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-ral 2711
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