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Theorem anim12 809
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 609. 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
anim12 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Proof of Theorem anim12
StepHypRef Expression
1 id 22 . 2 ((𝜑𝜓) → (𝜑𝜓))
2 id 22 . 2 ((𝜒𝜃) → (𝜒𝜃))
31, 2im2anan9 620 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395
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 207  df-an 396
This theorem is referenced by:  euind  3733  reuind  3762  reusv3i  5410  opelopabt  5542  wemaplem2  9585  rexanre  15382  rlimcn3  15623  o1of2  15646  o1rlimmul  15652  2sqlem6  27482  spanuni  31573  bj-nnfan  36731  isbasisrelowllem1  37338  isbasisrelowllem2  37339  heicant  37642  pm11.71  44393
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