ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exbi Unicode version

Theorem exbi 1584
Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
exbi  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )

Proof of Theorem exbi
StepHypRef Expression
1 bi1 117 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21alimi 1432 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ph  ->  ps ) )
3 exim 1579 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
42, 3syl 14 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  ->  E. x ps )
)
5 bi2 129 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
65alimi 1432 . . 3  |-  ( A. x ( ph  <->  ps )  ->  A. x ( ps 
->  ph ) )
7 exim 1579 . . 3  |-  ( A. x ( ps  ->  ph )  ->  ( E. x ps  ->  E. x ph ) )
86, 7syl 14 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ps 
->  E. x ph )
)
94, 8impbid 128 1  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330   E.wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  exbii  1585  exbidh  1594  exintrbi  1613  19.19  1645  rexrnmpt  5570
  Copyright terms: Public domain W3C validator