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Theorem prth 594
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 585. 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 473 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
2 simpr 477 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜒𝜃))
31, 2anim12d 585 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384
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 197  df-an 386
This theorem is referenced by:  euind  3380  reuind  3398  reusv3i  4840  opelopabt  4952  wemaplem2  8397  rexanre  14015  rlimcn2  14250  o1of2  14272  o1rlimmul  14278  2sqlem6  25043  spanuni  28243  bj-mo3OLD  32450  isbasisrelowllem1  32808  isbasisrelowllem2  32809  heicant  33043  pm11.71  38046
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