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Theorem prth 554
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 546. 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 443 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ph  ->  ps ) )
2 simpr 447 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ch  ->  th ) )
31, 2anim12d 546 1  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  (
( ph  /\  ch )  ->  ( ps  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358
This theorem is referenced by:  mo  2178  2mo  2234  euind  2965  reuind  2981  reuss2  3461  opelopabt  4293  reusv3i  4557  tfrlem5  6412  wemaplem2  7278  rexanre  11846  rlimcn2  12080  o1of2  12102  o1rlimmul  12108  2sqlem6  20624  spanuni  22139  pm11.71  27698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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