MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.30 Structured version   Unicode version

Theorem r19.30 2964
Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
Assertion
Ref Expression
r19.30  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )

Proof of Theorem r19.30
StepHypRef Expression
1 ralim 2810 . 2  |-  ( A. x  e.  A  ( -.  ps  ->  ph )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
2 orcom 387 . . . 4  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
3 df-or 370 . . . 4  |-  ( ( ps  \/  ph )  <->  ( -.  ps  ->  ph )
)
42, 3bitri 249 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ps  ->  ph )
)
54ralbii 2834 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ps  ->  ph ) )
6 orcom 387 . . 3  |-  ( ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
7 dfrex2 2850 . . . 4  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
87orbi2i 519 . . 3  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps ) )
9 imor 412 . . 3  |-  ( ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
106, 8, 93bitr4i 277 . 2  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
111, 5, 103imtr4i 266 1  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368   A.wral 2795   E.wrex 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-ral 2800  df-rex 2801
This theorem is referenced by:  disjunsn  26080  esumcvg  26673
  Copyright terms: Public domain W3C validator