Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nxtand Unicode version

Theorem nxtand 24986
Description:  (
ph  /\  ps ) holds in the next step iff  ph holds in the next step and  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
nxtand  |-  ( () ( ph  /\  ps ) 
<->  ( () ph  /\  () ps ) )

Proof of Theorem nxtand
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
21impxt 24983 . . 3  |-  ( () ( ph  /\  ps )  ->  () ph )
3 simpr 447 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
43impxt 24983 . . 3  |-  ( () ( ph  /\  ps )  ->  () ps )
52, 4jca 518 . 2  |-  ( () ( ph  /\  ps )  ->  ( () ph  /\  () ps ) )
6 pm3.2 434 . . . . 5  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
76impxt 24983 . . . 4  |-  ( ()
ph  ->  () ( ps 
->  ( ph  /\  ps ) ) )
8 ax-ltl3 24976 . . . 4  |-  ( () ( ps  ->  ( ph  /\  ps ) )  ->  ( () ps  ->  () ( ph  /\  ps ) ) )
97, 8syl 15 . . 3  |-  ( ()
ph  ->  ( () ps  ->  () ( ph  /\  ps ) ) )
109imp 418 . 2  |-  ( ( () ph  /\  () ps )  ->  () (
ph  /\  ps )
)
115, 10impbii 180 1  |-  ( () ( ph  /\  ps ) 
<->  ( () ph  /\  () ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   ()wcirc 24972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl3 24976  ax-nmp 24979
This theorem depends on definitions:  df-bi 177  df-an 360
  Copyright terms: Public domain W3C validator