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

Theorem albi 1686
Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
albi  |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  A. x ps ) )

Proof of Theorem albi
StepHypRef Expression
1 biimp 196 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
21al2imi 1683 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  ->  A. x ps )
)
3 biimpr 201 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
43al2imi 1683 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( A. x ps 
->  A. x ph )
)
52, 4impbid 193 1  |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  A. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188
This theorem is referenced by:  albii  1687  albidh  1720  19.16  2013  19.17  2014  equveli  2143  eqeq1d  2424  intmin4  4282  dfiin2g  4329  bj-2albi  31192  bj-hbxfrbi  31203  bj-nfbi  31204  wl-aleq  31782  2albi  36585  ralbidar  36656  sbcssOLD  36765  trsbcVD  37135  sbcssgVD  37141
  Copyright terms: Public domain W3C validator