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

Theorem prth 557
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 548. Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
Assertion
Ref Expression
prth  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  (
( ph  /\  ch )  ->  ( ps  /\  th ) ) )

Proof of Theorem prth
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ph  ->  ps ) )
2 simpr 449 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ch  ->  th ) )
31, 2anim12d 548 1  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  (
( ph  /\  ch )  ->  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360
This theorem is referenced by:  mo  2135  2mo  2191  euind  2889  reuind  2903  reuss2  3355  opelopabt  4170  reusv3i  4432  tfrlem5  6282  wemaplem2  7146  rexanre  11707  rlimcn2  11941  o1of2  11963  o1rlimmul  11969  2sqlem6  20440  spanuni  21953  pm11.71  26762
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