MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded9.20-2 Unicode version

Theorem ex-natded9.20-2 20758
Description: A more efficient proof of Theorem 9.20 of [Laboreo] p. 45. Compare with ex-natded9.20 20757. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
Hypothesis
Ref Expression
ex-natded9.20.1  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
Assertion
Ref Expression
ex-natded9.20-2  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )

Proof of Theorem ex-natded9.20-2
StepHypRef Expression
1 ex-natded9.20.1 . . . . 5  |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )
21simpld 447 . . . 4  |-  ( ph  ->  ps )
32anim1i 554 . . 3  |-  ( (
ph  /\  ch )  ->  ( ps  /\  ch ) )
43orcd 383 . 2  |-  ( (
ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
52anim1i 554 . . 3  |-  ( (
ph  /\  th )  ->  ( ps  /\  th ) )
65olcd 384 . 2  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
71simprd 451 . 2  |-  ( ph  ->  ( ch  \/  th ) )
84, 6, 7mpjaodan 764 1  |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ 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-or 361  df-an 362
  Copyright terms: Public domain W3C validator