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Mirrors > Home > MPE Home > Th. List > anim12 | Structured version Visualization version GIF version |
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 610. 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 621 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: euind 3715 reuind 3744 reusv3i 5305 opelopabt 5419 wemaplem2 9011 rexanre 14706 rlimcn2 14947 o1of2 14969 o1rlimmul 14975 2sqlem6 25999 spanuni 29321 bj-nnfan 34077 isbasisrelowllem1 34639 isbasisrelowllem2 34640 heicant 34942 pm11.71 40778 |
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