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Theorem prth 587
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 578. 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 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Proof of Theorem prth
StepHypRef Expression
1 simpl 466 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
2 simpr 470 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
31, 2anim12d 578 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 192  df-an 380
This theorem is referenced by:  euind  3249  reuind  3267  reuss2  3749  reusv3i  4649  opelopabt  4754  wemaplem2  8145  rexanre  13569  rlimcn2  13814  o1of2  13836  o1rlimmul  13842  2sqlem6  24458  spanuni  27360  bj-mo3OLD  31631  isbasisrelowllem1  31979  isbasisrelowllem2  31980  heicant  32213  pm11.71  37104
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