MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.32 Structured version   Visualization version   Unicode version

Theorem 19.32 2057
Description: Theorem 19.32 of [Margaris] p. 90. See 19.32v 1799 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypothesis
Ref Expression
19.32.1  |-  F/ x ph
Assertion
Ref Expression
19.32  |-  ( A. x ( ph  \/  ps )  <->  ( ph  \/  A. x ps ) )

Proof of Theorem 19.32
StepHypRef Expression
1 19.32.1 . . . 4  |-  F/ x ph
21nfn 1993 . . 3  |-  F/ x  -.  ph
3219.21 1997 . 2  |-  ( A. x ( -.  ph  ->  ps )  <->  ( -.  ph 
->  A. x ps )
)
4 df-or 376 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
54albii 1701 . 2  |-  ( A. x ( ph  \/  ps )  <->  A. x ( -. 
ph  ->  ps ) )
6 df-or 376 . 2  |-  ( (
ph  \/  A. x ps )  <->  ( -.  ph  ->  A. x ps )
)
73, 5, 63bitr4i 285 1  |-  ( A. x ( ph  \/  ps )  <->  ( ph  \/  A. x ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374   A.wal 1452   F/wnf 1677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by:  19.31  2058  2eu3  2394  axi12  2439
  Copyright terms: Public domain W3C validator