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| 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.) | 
| Ref | Expression | 
|---|---|
| anim12 | ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
| 2 | id 22 | . 2 ⊢ ((𝜒 → 𝜃) → (𝜒 → 𝜃)) | |
| 3 | 1, 2 | im2anan9 620 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | 
| 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 3729 reuind 3758 reusv3i 5403 opelopabt 5536 wemaplem2 9588 rexanre 15386 rlimcn3 15627 o1of2 15650 o1rlimmul 15656 2sqlem6 27468 spanuni 31564 bj-nnfan 36750 isbasisrelowllem1 37357 isbasisrelowllem2 37358 heicant 37663 pm11.71 44421 | 
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