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Theorem prth 593
 Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 584. 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 472 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
2 simpr 476 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
31, 2anim12d 584 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383 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 196  df-an 385 This theorem is referenced by:  euind  3360  reuind  3378  reusv3i  4801  opelopabt  4912  wemaplem2  8335  rexanre  13934  rlimcn2  14169  o1of2  14191  o1rlimmul  14197  2sqlem6  24948  spanuni  27787  bj-mo3OLD  32022  isbasisrelowllem1  32379  isbasisrelowllem2  32380  heicant  32614  pm11.71  37619
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