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Theorem prth 556
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  2167  2mo  2223  euind  2954  reuind  2970  reuss2  3450  opelopabt  4277  reusv3i  4541  tfrlem5  6392  wemaplem2  7258  rexanre  11825  rlimcn2  12059  o1of2  12081  o1rlimmul  12087  2sqlem6  20603  spanuni  22116  pm11.71  26996
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
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