ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prth Unicode version

Theorem prth 330
Description: 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 106 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ph  ->  ps ) )
2 simpr 107 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ch  ->  th ) )
31, 2anim12d 322 1  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  (
( ph  /\  ch )  ->  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  nfand  1476  equsexd  1633  mo23  1957  euind  2751  reuind  2767  reuss2  3245  opelopabt  4027  reusv3i  4219
  Copyright terms: Public domain W3C validator