MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded5.8-2 Unicode version

Theorem ex-natded5.8-2 20754
Description: A more efficient proof of Theorem 5.8 of [Laboreo] p. 20. For a longer line-by-line translation, see ex-natded5.8 20753. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
ex-natded5.8.1  |-  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) )
ex-natded5.8.2  |-  ( ph  ->  ( ta  ->  th )
)
ex-natded5.8.3  |-  ( ph  ->  ch )
ex-natded5.8.4  |-  ( ph  ->  ta )
Assertion
Ref Expression
ex-natded5.8-2  |-  ( ph  ->  -.  ps )

Proof of Theorem ex-natded5.8-2
StepHypRef Expression
1 ex-natded5.8.4 . . 3  |-  ( ph  ->  ta )
2 ex-natded5.8.2 . . 3  |-  ( ph  ->  ( ta  ->  th )
)
31, 2mpd 16 . 2  |-  ( ph  ->  th )
4 ex-natded5.8.3 . . 3  |-  ( ph  ->  ch )
5 ex-natded5.8.1 . . 3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) )
64, 5mpan2d 658 . 2  |-  ( ph  ->  ( ps  ->  -.  th ) )
73, 6mt2d 111 1  |-  ( ph  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362
  Copyright terms: Public domain W3C validator